Recent progress and open questions on the numerical index of ...
Recent progress and open questions on the numerical index of ...
Recent progress and open questions on the numerical index of ...
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RACSAMRev. R. Acad. Cien. Serie A. Mat.VOL. 2000 (1-2), 2006, pp. 155–182Análisis Matemático / Ma<strong>the</strong>matical AnalysisArtículo panorámico / Survey<str<strong>on</strong>g>Recent</str<strong>on</strong>g> <str<strong>on</strong>g>progress</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>open</str<strong>on</strong>g> <str<strong>on</strong>g>questi<strong>on</strong>s</str<strong>on</strong>g> <strong>on</strong> <strong>the</strong> <strong>numerical</strong> <strong>index</strong><strong>of</strong> Banach spacesVladimir Kadets, Miguel Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> Rafael PayáAbstract The aim <strong>of</strong> this paper is to review <strong>the</strong> state-<strong>of</strong>-<strong>the</strong>-art <strong>of</strong> recent research c<strong>on</strong>cerning <strong>the</strong> <strong>numerical</strong><strong>index</strong> <strong>of</strong> Banach spaces, by presenting some <strong>of</strong> <strong>the</strong> results found in <strong>the</strong> last years <str<strong>on</strong>g>and</str<strong>on</strong>g> proposing anumber <strong>of</strong> related <str<strong>on</strong>g>open</str<strong>on</strong>g> problems.Resultados recientes y problemas abiertos sobre el índice numérico de losespacios de BanachResumen. El propósito de este trabajo es revisar el estado actual de la investigación reciente sobreel índice numérico de los espacios de Banach, present<str<strong>on</strong>g>and</str<strong>on</strong>g>o algunos de los resultados obtenidos en losúltimos años y prop<strong>on</strong>iendo un cierto número de problemas abiertos.Introducti<strong>on</strong>The <strong>numerical</strong> <strong>index</strong> <strong>of</strong> a Banach space is a c<strong>on</strong>stant relating <strong>the</strong> behavior <strong>of</strong> <strong>the</strong> <strong>numerical</strong> range withthat <strong>of</strong> <strong>the</strong> usual norm <strong>on</strong> <strong>the</strong> Banach algebra <strong>of</strong> all bounded linear operators <strong>on</strong> <strong>the</strong> space. The noti<strong>on</strong> <strong>of</strong><strong>numerical</strong> range (also called field <strong>of</strong> values) was first introduced by O. Toeplitz in 1918 [79] for matrices,but his definiti<strong>on</strong> applies equally well to operators <strong>on</strong> infinite-dimensi<strong>on</strong>al Hilbert spaces. If H denotes aHilbert space with inner product (· | ·), <strong>the</strong> <strong>numerical</strong> range <strong>of</strong> a bounded linear operator T <strong>on</strong> H is <strong>the</strong>subset W (T ) <strong>of</strong> <strong>the</strong> scalar field defined byW (T ) := { (T x | x) : x ∈ H, (x | x) = 1 } .Some properties <strong>of</strong> <strong>the</strong> Hilbert space <strong>numerical</strong> range are discussed in <strong>the</strong> classical book <strong>of</strong> P. Halmos [29,§17]. Let us just menti<strong>on</strong> that <strong>the</strong> <strong>numerical</strong> range <strong>of</strong> a bounded linear operator is (surprisingly) c<strong>on</strong>vex <str<strong>on</strong>g>and</str<strong>on</strong>g>,in <strong>the</strong> complex case, its closure c<strong>on</strong>tains <strong>the</strong> spectrum <strong>of</strong> <strong>the</strong> operator. Moreover, if <strong>the</strong> operator is normal,<strong>the</strong>n <strong>the</strong> closure <strong>of</strong> its <strong>numerical</strong> range coincides with <strong>the</strong> c<strong>on</strong>vex hull <strong>of</strong> its spectrum. Fur<strong>the</strong>r developmentscan be found in a recent book <strong>of</strong> K. Gustafs<strong>on</strong> <str<strong>on</strong>g>and</str<strong>on</strong>g> D. Rao [28]. In <strong>the</strong> sixties, <strong>the</strong> c<strong>on</strong>cept <strong>of</strong> <strong>numerical</strong>range was extended to operators <strong>on</strong> general Banach spaces by G. Lumer [54] <str<strong>on</strong>g>and</str<strong>on</strong>g> F. Bauer [4]. Let us give<strong>the</strong> necessary definiti<strong>on</strong>s. Given a real or complex Banach space X, we write B X for <strong>the</strong> closed unit ball<str<strong>on</strong>g>and</str<strong>on</strong>g> S X for <strong>the</strong> unit sphere <strong>of</strong> X. The dual space will be denoted by X ∗ <str<strong>on</strong>g>and</str<strong>on</strong>g> L(X) will be <strong>the</strong> Banachalgebra <strong>of</strong> all bounded linear operators <strong>on</strong> X. The <strong>numerical</strong> range <strong>of</strong> an operator T ∈ L(X) is <strong>the</strong> subsetV (T ) <strong>of</strong> <strong>the</strong> scalar field defined byV (T ) := {x ∗ (T x) : x ∈ S X , x ∗ ∈ S X ∗, x ∗ (x) = 1}.Presentado por Vicente M<strong>on</strong>tesinos Santalucía.Recibido: 28/03/2006. Aceptado: 01/06/2006.Palabras clave / Keywords: Numerical range; <strong>numerical</strong> radius; <strong>numerical</strong> <strong>index</strong>; duality; Daugavet equati<strong>on</strong>.Ma<strong>the</strong>matics Subject Classificati<strong>on</strong>s: 46B20, 47A12.c○ 2006 Real Academia de Ciencias, España.155
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. PayáThe <strong>numerical</strong> radius is <strong>the</strong> seminorm defined <strong>on</strong> L(X) byv(T ) := sup{|λ| : λ ∈ V (T )}for T ∈ L(X). Classical references here are <strong>the</strong> m<strong>on</strong>ographs by F. B<strong>on</strong>sall <str<strong>on</strong>g>and</str<strong>on</strong>g> J. Duncan [9, 10] from<strong>the</strong> seventies. Let us menti<strong>on</strong> that <strong>the</strong> <strong>numerical</strong> range <strong>of</strong> a bounded linear operator is c<strong>on</strong>nected (but notnecessarily c<strong>on</strong>vex, see [10, Example 21.6]) <str<strong>on</strong>g>and</str<strong>on</strong>g>, in <strong>the</strong> complex case, its closure c<strong>on</strong>tains <strong>the</strong> spectrum<strong>of</strong> <strong>the</strong> operator. The <strong>the</strong>ory <strong>of</strong> <strong>numerical</strong> ranges has played a crucial role in <strong>the</strong> study <strong>of</strong> some algebraicstructures, especially in <strong>the</strong> n<strong>on</strong>-associative c<strong>on</strong>text (see <strong>the</strong> expository paper [46] by A. Kaidi, A. Morales,<str<strong>on</strong>g>and</str<strong>on</strong>g> A. Rodríguez Palacios, for example).The c<strong>on</strong>cept <strong>of</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> a Banach space X was first suggested by G. Lumer in 1968 (see[20]), <str<strong>on</strong>g>and</str<strong>on</strong>g> it is <strong>the</strong> c<strong>on</strong>stant n(X) defined byn(X) := inf{v(T ) : T ∈ L(X), ‖T ‖ = 1}or, equivalently,n(X) = max{k 0 : k ‖T ‖ v(T ) ∀ T ∈ L(X)}.Note that n(X) > 0 if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if v <str<strong>on</strong>g>and</str<strong>on</strong>g> ‖ · ‖ are equivalent norms <strong>on</strong> L(X). At that time, it was knownthat in a complex Hilbert space H with dimensi<strong>on</strong> greater than 1, ‖T ‖ 2v(T ) for all T ∈ L(H), <str<strong>on</strong>g>and</str<strong>on</strong>g> 2 is<strong>the</strong> best c<strong>on</strong>stant; in <strong>the</strong> real case, <strong>the</strong>re exists a norm-<strong>on</strong>e operator whose <strong>numerical</strong> range reduces to zero.In our terminology, n(H) = 1/2 if H is complex, <str<strong>on</strong>g>and</str<strong>on</strong>g> n(H) = 0 if it is real. Actually, real <str<strong>on</strong>g>and</str<strong>on</strong>g> complexgeneral Banach spaces behave in a very different way with regard to <strong>the</strong> <strong>numerical</strong> <strong>index</strong>, as summarized in<strong>the</strong> following equalities by J. Duncan, C. McGregor, J. Pryce, <str<strong>on</strong>g>and</str<strong>on</strong>g> A. White [20]:{n(X) : X complex Banach space } = [e −1 , 1], (1){n(X) : X real Banach space } = [0, 1].The fact that n(X) e −1 for every complex Banach space X was observed by B. Glickfeld [26] (bymaking use <strong>of</strong> a classical <strong>the</strong>orem <strong>of</strong> H. Bohnenblust <str<strong>on</strong>g>and</str<strong>on</strong>g> S. Karlin [7]), who also gave an example wherethis inequality becomes an equality. It is showed in <strong>the</strong> already cited paper [20], that M-spaces, L-spaces<str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong>ir isometric preduals, have <strong>numerical</strong> <strong>index</strong> 1, a property shared by <strong>the</strong> disk algebra (M. Crab,J. Duncan, <str<strong>on</strong>g>and</str<strong>on</strong>g> C. McGregor [16, Theorem 3.3]). Finally, let us menti<strong>on</strong> that <strong>the</strong> real space X R underlyinga complex Banach space X satisfies n(X R ) = 0; actually, <strong>the</strong> isometry x ↦−→ i x has <strong>numerical</strong> radius 0when viewed as an operator <strong>on</strong> X R .In <strong>the</strong> last ten years, many results <strong>on</strong> <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> Banach spaces have appeared in <strong>the</strong> literature.This paper aims at reviewing <strong>the</strong> state <strong>of</strong> <strong>the</strong> art <strong>on</strong> this topic <str<strong>on</strong>g>and</str<strong>on</strong>g> proposing a variety <strong>of</strong> <str<strong>on</strong>g>open</str<strong>on</strong>g> <str<strong>on</strong>g>questi<strong>on</strong>s</str<strong>on</strong>g>.The structure <strong>of</strong> our discussi<strong>on</strong> is as follows.C<strong>on</strong>tents1. Computing <strong>the</strong> <strong>numerical</strong> <strong>index</strong> 1572. Numerical <strong>index</strong> <str<strong>on</strong>g>and</str<strong>on</strong>g> duality 1603. Banach spaces with <strong>numerical</strong> <strong>index</strong> <strong>on</strong>e 1634. Renorming <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>numerical</strong> <strong>index</strong> 1665. Real Banach spaces with <strong>numerical</strong> <strong>index</strong> zero 1686. Asymptotic behavior <strong>of</strong> <strong>the</strong> set <strong>of</strong> finite-dimensi<strong>on</strong>al spaces with <strong>numerical</strong> <strong>index</strong> <strong>on</strong>e 171156
Numerical <strong>index</strong> <strong>of</strong> Banach spaces7. Relati<strong>on</strong>ship to <strong>the</strong> Daugavet property. 1728. The polynomial <strong>numerical</strong> indices 175We finish this introducti<strong>on</strong> by recalling some definiti<strong>on</strong>s <str<strong>on</strong>g>and</str<strong>on</strong>g> fixing notati<strong>on</strong>. We write T to denote <strong>the</strong>unit sphere <strong>of</strong> <strong>the</strong> base field K (= R or C). We use <strong>the</strong> notati<strong>on</strong> Re( · ) to denote <strong>the</strong> real part functi<strong>on</strong>,which should be c<strong>on</strong>sidered as <strong>the</strong> identity when K = R. Given a real or complex Banach space X, wewrite co(B) <str<strong>on</strong>g>and</str<strong>on</strong>g> co(B) to denote, respectively, <strong>the</strong> c<strong>on</strong>vex hull <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> closed c<strong>on</strong>vex hull <strong>of</strong> a set B ⊆ X<str<strong>on</strong>g>and</str<strong>on</strong>g> we denote by ext(C) <strong>the</strong> set <strong>of</strong> extreme points <strong>of</strong> a c<strong>on</strong>vex set C ⊆ X. A subset A <strong>of</strong> B X ∗ is said tobe norming (for X) if‖x‖ = sup{|x ∗ (x)| : x ∗ ( )∈ A} x ∈ Xor, equivalently, if B X ∗ = co w∗ (T A) (Hahn-Banach Theorem). Finally, for 1 p ∞, we write l m p todenote <strong>the</strong> normed space K m endowed with <strong>the</strong> usual p-norm, <str<strong>on</strong>g>and</str<strong>on</strong>g> we write X ⊕ p Y to denote <strong>the</strong> l p -directsum <strong>of</strong> <strong>the</strong> spaces X <str<strong>on</strong>g>and</str<strong>on</strong>g> Y .1. Computing <strong>the</strong> <strong>numerical</strong> <strong>index</strong>In view <strong>of</strong> <strong>the</strong> examples given in <strong>the</strong> introducti<strong>on</strong>, <strong>the</strong> most important family <strong>of</strong> classical Banach spaces(in <strong>the</strong> sense <strong>of</strong> H. Lacey [47]) whose <strong>numerical</strong> indices remain unknown is <strong>the</strong> family <strong>of</strong> L p spaces whenp ≠ 1, 2, ∞. This is actually <strong>on</strong>e <strong>of</strong> <strong>the</strong> most intriguing <str<strong>on</strong>g>open</str<strong>on</strong>g> problems in <strong>the</strong> field but, very recently,E. Ed-Dari <str<strong>on</strong>g>and</str<strong>on</strong>g> M. Khamsi [21, 22] have made some <str<strong>on</strong>g>progress</str<strong>on</strong>g>. We summarize <strong>the</strong>ir results in <strong>the</strong> followingstatement <str<strong>on</strong>g>and</str<strong>on</strong>g> use it to motivate some c<strong>on</strong>jectures.Theorem 1 ([21, 22]) Let 1 p ∞ be fixed. Then,(a) n(L p [0, 1]) = n(l p ) = inf{n(l m p ) : m ∈ N}, <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> sequence {n(l m p )} m∈N is decreasing.(b) n(L p (µ)) n(l p ) for every positive measure µ.(c) In <strong>the</strong> real case,12 M p n(l 2 |t p−1 − t|p) M p , where M p = supt∈[0,1] 1 + t p .When p ≠ 2, it is known that n(l m p ) > 0 for every m 2, <str<strong>on</strong>g>and</str<strong>on</strong>g> also that v(T ) > 0 for every n<strong>on</strong>-nullT ∈ L(l p ) (see [22, Theorem 2.3 <str<strong>on</strong>g>and</str<strong>on</strong>g> subsequent remark]), but we do not know if n(l p ) > 0. Observethat a positive answer to this questi<strong>on</strong> implies, thanks to (b) above, that n(L p (µ)) > 0 for every positivemeasure µ.Problem 1 Is n(l p ) positive for every p ≠ 2?With respect to item (c) in <strong>the</strong> above <strong>the</strong>orem, let us explain <strong>the</strong> meaning <strong>of</strong> <strong>the</strong>(number)M p . It can bea bdeduced from [20, §3] that, given an operator T ∈ L(l 2 p) represented by <strong>the</strong> matrix , <strong>on</strong>e hasc d⎧⎪⎨∣∣a + d t p + z b t + z c t p−1∣ ∣∣ ∣d + a t p + z c t + z b t p−1∣ ∣v(T ) = max⎪⎩ maxt∈[0,1]z∈T1 + t p , maxt∈[0,1]z∈T1 + t p ⎫⎪⎬⎪ ⎭. (2)( )0 1It follows that M p is equal to <strong>the</strong> <strong>numerical</strong> radius <strong>of</strong> <strong>the</strong> norm-<strong>on</strong>e operator U ≡ in L(l−1 02 p) (realcase), so n(l 2 p) M p . For p = 2, <strong>the</strong> operator U has minimum <strong>numerical</strong> radius, namely 0. We may ask ifU is also <strong>the</strong> norm-<strong>on</strong>e operator with minimum <strong>numerical</strong> radius for all <strong>the</strong> real spaces l 2 p.157
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. PayáProblem 2 Is it true that, in <strong>the</strong> real case, n(l 2 |t p−1 − t|p) = supt∈[0,1] 1 + t p for every 1 < p < ∞?In <strong>the</strong> complex case, <strong>the</strong> operator U acting <strong>on</strong> l 2 2 satisfies v(U) = ‖U‖ (take z = i <str<strong>on</strong>g>and</str<strong>on</strong>g> t = 1 in (2))<str<strong>on</strong>g>and</str<strong>on</strong>g>, <strong>the</strong>refore, its <strong>numerical</strong> radius is not <strong>the</strong> minimum. Actually, <strong>on</strong>e hasn(l 2 2) = 1 2 = v(S),( ) 0 0where S ∈ L(l 2 2) is <strong>the</strong> ‘shift’ S ≡ . Therefore, we bet that n(l1 02 p) = v(S) for every p in <strong>the</strong>complex case. It can be checked from (2) thatv(S) =p−1(p − 1)pp= 1p 1 1,p qqwhere 1 p + 1 q = 1.Problem 3 Is it true that, in <strong>the</strong> complex case, n(l 2 p) = 1p 1 p q1qfor every 1 < p < ∞?In view <strong>of</strong> Theorem 1.a, <strong>the</strong> two-dimensi<strong>on</strong>al case is <strong>on</strong>ly <strong>the</strong> first step in <strong>the</strong> computati<strong>on</strong> <strong>of</strong> n(l p ),but it is reas<strong>on</strong>able to expect that <strong>the</strong> sequence {n(l m p )} m∈N is always c<strong>on</strong>stant, as it happens in <strong>the</strong> casesp = 1, 2, ∞.Problem 4 Is it true that n(l p ) = n(l 2 p) for every 1 < p < ∞?In a 1977 paper [35], T. Huruya determined <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> a C ∗ -algebra. Part <strong>of</strong> <strong>the</strong> pro<strong>of</strong> wasrecently clarified by A. Kaidi, A. Morales, <str<strong>on</strong>g>and</str<strong>on</strong>g> A. Rodríguez-Palacios in [45], where <strong>the</strong> result is extendedto JB ∗ -algebras <str<strong>on</strong>g>and</str<strong>on</strong>g> preduals <strong>of</strong> JBW ∗ -algebras. Let us state here those results just for C ∗ -algebras <str<strong>on</strong>g>and</str<strong>on</strong>g>preduals <strong>of</strong> v<strong>on</strong> Neumann algebras.Theorem 2 ([35] <str<strong>on</strong>g>and</str<strong>on</strong>g> [45, Propositi<strong>on</strong> 2.8]) Let A be a C ∗ -algebra. Then, n(A) is equal to 1 or 1 2 depending<strong>on</strong> whe<strong>the</strong>r A is commutative or not. If A is actually a v<strong>on</strong> Neumann algebra with predual A ∗ ,<strong>the</strong>n n(A ∗ ) = n(A).We do not know if <strong>the</strong>re is an analogous result in <strong>the</strong> real case. We recall that a real C ∗ -algebra can bedefined as a norm-closed self-adjoint real subalgebra <strong>of</strong> a complex C ∗ -algebra, <str<strong>on</strong>g>and</str<strong>on</strong>g> a real W ∗ -algebra (orreal v<strong>on</strong> Neumann algebra) is a real C ∗ -algebra which admits a predual (see [36] for more informati<strong>on</strong>).Problem 5 Compute <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> real C ∗ -algebras <str<strong>on</strong>g>and</str<strong>on</strong>g> isometric preduals <strong>of</strong> real W ∗ -algebras.The fact that <strong>the</strong> disk algebra has <strong>numerical</strong> <strong>index</strong> 1 was extended to functi<strong>on</strong> algebras by D. Werner in1997 [82]. A functi<strong>on</strong> algebra A <strong>on</strong> a compact Hausdorff space K is a closed subalgebra <strong>of</strong> C(K) whichseparates <strong>the</strong> points <strong>of</strong> K <str<strong>on</strong>g>and</str<strong>on</strong>g> c<strong>on</strong>tains <strong>the</strong> c<strong>on</strong>stant functi<strong>on</strong>s.Propositi<strong>on</strong> 1 ([82, Corollary 2.2 <str<strong>on</strong>g>and</str<strong>on</strong>g> pro<strong>of</strong> <strong>of</strong> Theorem 3.3]) If A is a functi<strong>on</strong> algebra, <strong>the</strong>n n(A) = 1.Of course, <strong>the</strong>re are many o<strong>the</strong>r Banach spaces whose <strong>numerical</strong> <strong>index</strong> is unknown. We propose tocalculate some <strong>of</strong> <strong>the</strong>m.Problem 6 Compute <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> C m [0, 1] (<strong>the</strong> space <strong>of</strong> m-times c<strong>on</strong>tinuously differentiablereal functi<strong>on</strong>s <strong>on</strong> [0, 1], endowed with any <strong>of</strong> its usual norms), Lip(K) (<strong>the</strong> space <strong>of</strong> all Lipschitz functi<strong>on</strong>s<strong>on</strong> <strong>the</strong> complete metric space K), Lorentz spaces, <str<strong>on</strong>g>and</str<strong>on</strong>g> Orlicz spaces.158
Numerical <strong>index</strong> <strong>of</strong> Banach spacesSome <strong>of</strong> <strong>the</strong> classical results given in <strong>the</strong> introducti<strong>on</strong> about <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> particular spaceshave been extended to sums <strong>of</strong> families <strong>of</strong> Banach spaces <str<strong>on</strong>g>and</str<strong>on</strong>g> to spaces <strong>of</strong> vector-valued functi<strong>on</strong>s in variouspapers by G. López, M. Martín, J. Merí, R. Payá, <str<strong>on</strong>g>and</str<strong>on</strong>g> A. Villena [52, 62, 64].We start by presenting <strong>the</strong> result for sums <strong>of</strong> spaces. Given an arbitrary family {X λ : λ ∈ Λ} <strong>of</strong>Banach spaces, we denote by [⊕ λ∈Λ X λ ] c0, [⊕ λ∈Λ X λ ] l1<str<strong>on</strong>g>and</str<strong>on</strong>g> [⊕ λ∈Λ X λ ] l∞<strong>the</strong> c 0 -, l 1 - <str<strong>on</strong>g>and</str<strong>on</strong>g> l ∞ -sum <strong>of</strong> <strong>the</strong>family.Propositi<strong>on</strong> 2 ([62, Propositi<strong>on</strong> 1]) Let {X λ : λ ∈ Λ} be a family <strong>of</strong> Banach spaces. Then)))n([⊕ λ∈Λ X λ ] c0= n([⊕ λ∈Λ X λ ] l1= n([⊕ λ∈Λ X λ ] l∞= inf n(X λ).λThe above result is not true for l p -sums if p is different from 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> ∞. Never<strong>the</strong>less, it is possible togive <strong>on</strong>e inequality <str<strong>on</strong>g>and</str<strong>on</strong>g>, actually, <strong>the</strong> same is true for absolute sums [56]. Recall that a direct sum Y ⊕ Z issaid to be an absolute sum if ‖y + z‖ <strong>on</strong>ly depends <strong>on</strong> ‖y‖ <str<strong>on</strong>g>and</str<strong>on</strong>g> ‖z‖ for (y, z) ∈ Y × Z. For background <strong>on</strong>absolute sums <strong>the</strong> reader is referred to [66] <str<strong>on</strong>g>and</str<strong>on</strong>g> references <strong>the</strong>rein.Propositi<strong>on</strong> 3 ([56, Proposición 1]) Let X be a Banach space <str<strong>on</strong>g>and</str<strong>on</strong>g> let Y , Z be closed subspaces <strong>of</strong> X.Suppose that X is <strong>the</strong> absolute sum <strong>of</strong> Y <str<strong>on</strong>g>and</str<strong>on</strong>g> Z. Thenn(X) min { n(Y ), n(Z) } .The following somehow surprising example was obtained in [62] by using Propositi<strong>on</strong> 2.Example 1 ([62, Example 2.b]) There is a real Banach space X for which <strong>the</strong> <strong>numerical</strong> radius is a normbut is not equivalent to <strong>the</strong> operator norm, i.e. <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> X is 0 although v(T ) > 0 for everyn<strong>on</strong>-null T ∈ L(X).The <strong>numerical</strong> <strong>index</strong> <strong>of</strong> some vector-valued functi<strong>on</strong> spaces was also computed in [52, 62, 64]. Givena real or complex Banach space X <str<strong>on</strong>g>and</str<strong>on</strong>g> a compact Hausdorff topological space K, we write C(K, X) <str<strong>on</strong>g>and</str<strong>on</strong>g>C w (K, X) to denote, respectively, <strong>the</strong> space <strong>of</strong> X-valued c<strong>on</strong>tinuous (resp. weakly c<strong>on</strong>tinuous) functi<strong>on</strong>s<strong>on</strong> K. If µ is a positive σ-finite measure, by L 1 (µ, X) <str<strong>on</strong>g>and</str<strong>on</strong>g> L ∞ (µ, X) we denote respectively <strong>the</strong> space<strong>of</strong> X-valued µ-Bochner-integrable functi<strong>on</strong>s <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> space <strong>of</strong> X-valued µ-Bochner-measurable functi<strong>on</strong>swhich are essentially bounded.Theorem 3 ([52], [62], <str<strong>on</strong>g>and</str<strong>on</strong>g> [64]) Let K be a compact Hausdorff space, <str<strong>on</strong>g>and</str<strong>on</strong>g> let µ be a positive σ-finitemeasure. Thenfor every Banach space X.n(C w (K, X)) = n(C(K, X)) = n(L 1 (µ, X)) = n(L ∞ (µ, X)) = n(X)The <strong>numerical</strong> <strong>index</strong> <strong>of</strong> C w ∗(K, X ∗ ), <strong>the</strong> space <strong>of</strong> X ∗ -valued weakly-star c<strong>on</strong>tinuous functi<strong>on</strong>s <strong>on</strong> Kis also studied in [52]. Unfortunately, <strong>on</strong>ly a partial result is achieved.Propositi<strong>on</strong> 4 ([52, Propositi<strong>on</strong>s 5 <str<strong>on</strong>g>and</str<strong>on</strong>g> 7] ) Let K be a compact Hausdorff space <str<strong>on</strong>g>and</str<strong>on</strong>g> let X be a Banachspace. Thenn(C w ∗(K, X ∗ )) n(X).If, in additi<strong>on</strong>, X is an Asplund space or K has a dense subset <strong>of</strong> isolated points, <strong>the</strong>nn(X ∗ ) n(C w ∗(K, X ∗ )).159
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. PayáTo finish this secti<strong>on</strong> let us comment that, roughly speaking, when <strong>on</strong>e finds an explicit computati<strong>on</strong><strong>of</strong> <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> a Banach space in <strong>the</strong> literature <strong>on</strong>ly few values appear; namely, 0 (real Hilbertspaces), e −1 (Glickfeld’s example), 1/2 (complex Hilbert spaces), <str<strong>on</strong>g>and</str<strong>on</strong>g> 1 (C(K), L 1 (µ), <str<strong>on</strong>g>and</str<strong>on</strong>g> many more).The preceding results about sums <str<strong>on</strong>g>and</str<strong>on</strong>g> vector-valued functi<strong>on</strong> spaces do not help so much, <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> exactvalues <strong>of</strong> n(l 2 p) are not still known. Let us also say that, when <strong>the</strong> authors <strong>of</strong> [20] prove (1), <strong>the</strong>y <strong>on</strong>lyuse examples <strong>of</strong> Banach spaces whose <strong>numerical</strong> indices are <strong>the</strong> extremes <strong>of</strong> <strong>the</strong> intervals, <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong>n ac<strong>on</strong>nectedness argument is applied. Very recently, M. Martín <str<strong>on</strong>g>and</str<strong>on</strong>g> J. Merí have partially covered this gapin [59], where <strong>the</strong>y explicitly compute <strong>the</strong> <strong>numerical</strong> <strong>index</strong> for four families <strong>of</strong> norms <strong>on</strong> R 2 . The mostinteresting <strong>on</strong>e is <strong>the</strong> family <strong>of</strong> regular polyg<strong>on</strong>s.Propositi<strong>on</strong> 5 ([59, Theorem 5]) Let n 2 be a positive integer, <str<strong>on</strong>g>and</str<strong>on</strong>g> let X n be <strong>the</strong> two-dimensi<strong>on</strong>alreal normed space whose unit ball is <strong>the</strong> c<strong>on</strong>vex hull <strong>of</strong> <strong>the</strong> (2n) th roots <strong>of</strong> unity, i.e. B Xn is a regular2n-polyg<strong>on</strong> centered at <strong>the</strong> origin <str<strong>on</strong>g>and</str<strong>on</strong>g> such that <strong>on</strong>e <strong>of</strong> its vertices is (1, 0). Then,⎧ ( πtan if n is even,⎪⎨ 2n)n(X n ) =(⎪⎩ πsin if n is odd.2n)Problem 7 Compute <strong>the</strong> <strong>numerical</strong> <strong>index</strong> for o<strong>the</strong>r families <strong>of</strong> finite-dimensi<strong>on</strong>al Banach spaces. In particular,it would be interesting to get <strong>the</strong> complex analog <strong>of</strong> <strong>the</strong> results in [59].2. Numerical <strong>index</strong> <str<strong>on</strong>g>and</str<strong>on</strong>g> dualityGiven a bounded linear operator T <strong>on</strong> a Banach space X, it is a well-known fact in <strong>the</strong> <strong>the</strong>ory <strong>of</strong> <strong>numerical</strong>ranges (see [9, §9]) that‖Id + α T ‖ − 1sup Re V (T ) = lim(3)α↓0 α<str<strong>on</strong>g>and</str<strong>on</strong>g> soIt is immediate from <strong>the</strong> above formula thatv(T ) = maxω∈T limα↓0‖Id + α ω T ‖ − 1.αv(T ) = v(T ∗ ), (4)where T ∗ ∈ L(X ∗ ) is <strong>the</strong> adjoint operator <strong>of</strong> T , <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> result given in [20, Propositi<strong>on</strong> 1.3] thatn(X ∗ ) n(X) (5)clearly follows. The questi<strong>on</strong> if this is actually an equality had been around from <strong>the</strong> beginning <strong>of</strong> <strong>the</strong>subject (see [46, pp. 386], for instance). Let us comment some partial results which led to think that <strong>the</strong>answer could be positive. Namely, it is clear that n(X) = n(X ∗ ) for every reflexive space X, <str<strong>on</strong>g>and</str<strong>on</strong>g> thisequality also holds whenever n(X ∗ ) = 1 , in particular when X is an L- or an M-space. Moreover, it isalso true that n(X) = n(X ∗ ) when X is a C ∗ -algebra or a v<strong>on</strong> Neumann algebra predual (Theorem 2).Never<strong>the</strong>less, in a very recent paper [12], K. Boyko, V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> D. Werner have answered<strong>the</strong> questi<strong>on</strong> in <strong>the</strong> negative by giving an example <strong>of</strong> a Banach space whose <strong>numerical</strong> <strong>index</strong> is strictlygreater than <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> its dual. Our aim in this secti<strong>on</strong> is to present such counterexample witha new <str<strong>on</strong>g>and</str<strong>on</strong>g> more direct pro<strong>of</strong>.Let us recall that c denotes <strong>the</strong> Banach space <strong>of</strong> all c<strong>on</strong>vergent scalar sequences x = (x(1), x(2), . . .)equipped with <strong>the</strong> sup-norm. The dual space <strong>of</strong> c is (isometric to) l 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> we will write c ∗ ≡ l 1 ⊕ 1 K where∞∑(〈(y, λ) , x〉 = y(n) x(n) + λ lim x x ∈ c, (y, λ) ∈ l1 ⊕ 1 K ) .160n=1
Numerical <strong>index</strong> <strong>of</strong> Banach spacesFor every n ∈ N, we denote by e ∗ n <strong>the</strong> norm-<strong>on</strong>e element <strong>of</strong> c ∗ given bye ∗ n(x) = x(n)(x ∈ c).We are now ready to show that <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> a Banach space <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> <strong>on</strong>e <strong>of</strong> its dual do not alwayscoincide.Example 2 ([12, Example 3.1]) Let us c<strong>on</strong>sider <strong>the</strong> Banach spaceThen, n(X) = 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> n(X ∗ ) < 1.X = { (x, y, z) ∈ c ⊕ ∞ c ⊕ ∞ c : lim x + lim y + lim z = 0 } .PROOF.We observe thatso that, writing Z = l 3 1/〈(1, 1, 1)〉, we can identifyX ∗ = [ c ∗ ⊕ 1 c ∗ ⊕ 1 c ∗] /〈(lim, lim, lim)〉X ∗ ≡ l 1 ⊕ 1 l 1 ⊕ 1 l 1 ⊕ 1 Z <str<strong>on</strong>g>and</str<strong>on</strong>g> X ∗∗ ≡ l ∞ ⊕ ∞ l ∞ ⊕ ∞ l ∞ ⊕ ∞ Z ∗ . (6)Figure 1. The unit ball <strong>of</strong> ZWith this in mind, we write A to denote <strong>the</strong> set{(e ∗ n, 0, 0, 0) : n ∈ N} ∪ {(0, e ∗ n, 0, 0) : n ∈ N} ∪ {(0, 0, e ∗ n, 0) : n ∈ N}.Then A is clearly a norming subset <strong>of</strong> S X ∗ <str<strong>on</strong>g>and</str<strong>on</strong>g>|x ∗∗ (a ∗ )| = 1(x ∗∗ ∈ ext(B X ∗∗), a ∗ ∈ A ) . (7)Let us prove that n(X) = 1. Indeed, we fix T ∈ L(X) <str<strong>on</strong>g>and</str<strong>on</strong>g> ε > 0. Since T ∗ is w ∗ -c<strong>on</strong>tinuous <str<strong>on</strong>g>and</str<strong>on</strong>g> A isnorming, we may find a ∗ ∈ A such thatNow, we take x ∗∗ ∈ ext(B X ∗∗) such thatSince |x ∗∗ (a ∗ )| = 1 thanks to (7), we get‖T ∗ (a ∗ )‖ ‖T ‖ − ε.|x ∗∗ (T ∗ (a ∗ ))| = ‖T ∗ (a ∗ )‖.v(T ) = v(T ∗ ) |x ∗∗ (T ∗ (a ∗ ))| ‖T ‖ − ε.161
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. PayáIt clearly follows that v(T ) = ‖T ‖ <str<strong>on</strong>g>and</str<strong>on</strong>g> n(X) = 1.To show that n(X ∗ ) < 1, we use (6) <str<strong>on</strong>g>and</str<strong>on</strong>g> Propositi<strong>on</strong> 2 to getn(X ∗ ) n(Z),<str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> fact that n(Z) < 1 follows easily from a result due to C. McGregor [65, Theorem 3.1]. Actually,in <strong>the</strong> real case, <strong>the</strong> unit ball <strong>of</strong> Z is an hexag<strong>on</strong> (see Figure 1 above), which is isometrically isomorphic to<strong>the</strong> space X 3 <strong>of</strong> Propositi<strong>on</strong> 5, so n(Z) = 1/2. The above example can be pushed forward, to produce even more striking counterexamples.Examples 3 ([12, Examples 3.3])(a) There exists a real Banach space X such that n(X) = 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> n(X ∗ ) = 0.(b) There exists a complex Banach space X satisfying that n(X) = 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> n(X ∗ ) = 1/e.Once we know that <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> a Banach space <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> <strong>on</strong>e <strong>of</strong> its dual may be different, <strong>the</strong>questi<strong>on</strong> aries if two preduals <strong>of</strong> a given Banach space have <strong>the</strong> same <strong>numerical</strong> <strong>index</strong>. By a predual <strong>of</strong> aBanach space Y we mean a Banach space X such that X ∗ is (isometrically isomorphic to) Y . The answeris again negative, as <strong>the</strong> following result shows.Example 4 ([12, Example 3.6]) Let us c<strong>on</strong>sider <strong>the</strong> Banach spaces<str<strong>on</strong>g>and</str<strong>on</strong>g>X 1 = { (x, y, z) ∈ c ⊕ ∞ c ⊕ ∞ c : lim x + lim y + lim z = 0 }X 2 = { (x, y, z) ∈ c ⊕ ∞ c ⊕ ∞ c : x(1) + y(1) + z(1) = 0 } .Then, X ∗ 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> X ∗ 2 are isometrically isomorphic, but n(X 1 ) = 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> n(X 2 ) < 1.The following questi<strong>on</strong> might also be addressed.Problem 8 Let Y be a dual space. Does <strong>the</strong>re exist a predual X <strong>of</strong> Y such that n(X) = n(Y )?Ano<strong>the</strong>r interesting issue could be to find isomorphic properties <strong>of</strong> a Banach space X ensuring thatn(X ∗ ) = n(X). On <strong>the</strong> <strong>on</strong>e h<str<strong>on</strong>g>and</str<strong>on</strong>g>, Example 2 shows that Asplundness is not such a property. On <strong>the</strong> o<strong>the</strong>rh<str<strong>on</strong>g>and</str<strong>on</strong>g>, it is shown in [12, Propositi<strong>on</strong> 4.1] that if a Banach space X with <strong>the</strong> Rad<strong>on</strong>-Nikodým property has<strong>numerical</strong> <strong>index</strong> 1, <strong>the</strong>n X ∗ has <strong>numerical</strong> <strong>index</strong> 1 as well. Therefore, <strong>the</strong> following questi<strong>on</strong> naturallyarises.Problem 9 Let X be a Banach space with <strong>the</strong> Rad<strong>on</strong>-Nikodým property. Is it true that n(X) = n(X ∗ )?Ano<strong>the</strong>r sufficient c<strong>on</strong>diti<strong>on</strong> would follow from a positive answer to Problem 8.Problem 10 Let Y be a dual space admitting a unique predual X (up to isometric isomorphisms). Is ittrue that n(Y ) = n(X)?Let us finish this secti<strong>on</strong> by remarking that <strong>the</strong> space given in Example 2 is useful as a counterexamplefor many o<strong>the</strong>r c<strong>on</strong>jectures, as we will see later <strong>on</strong>.162
Numerical <strong>index</strong> <strong>of</strong> Banach spaces3. Banach spaces with <strong>numerical</strong> <strong>index</strong> <strong>on</strong>eThe guiding <str<strong>on</strong>g>open</str<strong>on</strong>g> questi<strong>on</strong> <strong>on</strong> <strong>the</strong>se spaces is <strong>the</strong> following.Problem 11 Find necessary <str<strong>on</strong>g>and</str<strong>on</strong>g> sufficient c<strong>on</strong>diti<strong>on</strong>s for a Banach space to have <strong>numerical</strong> <strong>index</strong> 1 whichdo not involve operators.In 1971, C. McGregor [65, Theorem 3.1] gave such a characterizati<strong>on</strong> in <strong>the</strong> finite-dimensi<strong>on</strong>al case.More c<strong>on</strong>cretely, a finite-dimensi<strong>on</strong>al normed space X has <strong>numerical</strong> <strong>index</strong> 1 if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if|x ∗ (x)| = 1 for every x ∈ ext(B X ) <str<strong>on</strong>g>and</str<strong>on</strong>g> every x ∗ ∈ ext(B X ∗). (8)It is not clear how to extend this result to arbitrary Banach spaces. If we use literally (8) in <strong>the</strong> infinitedimensi<strong>on</strong>alc<strong>on</strong>text, we do not get a sufficient c<strong>on</strong>diti<strong>on</strong>, since <strong>the</strong> set ext(B X ) may be empty <str<strong>on</strong>g>and</str<strong>on</strong>g> thisdoes not imply <strong>numerical</strong> <strong>index</strong> 1 (e.g. ext(B c0 (l 2 )) = ∅ but n(c 0 (l 2 )) < 1). On <strong>the</strong> o<strong>the</strong>r h<str<strong>on</strong>g>and</str<strong>on</strong>g>, we do notknow if (8) is a necessary c<strong>on</strong>diti<strong>on</strong>.Problem 12 Let X be a Banach space with <strong>numerical</strong> <strong>index</strong> 1. Is it true that |x ∗ (x)| = 1 for everyx ∗ ∈ ext(B X ∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every x ∈ ext(B X )?Our first aim in this secti<strong>on</strong> is to discuss several reformulati<strong>on</strong>s <strong>of</strong> asserti<strong>on</strong> (8) to get ei<strong>the</strong>r sufficientor necessary c<strong>on</strong>diti<strong>on</strong>s for a Banach space to have <strong>numerical</strong> <strong>index</strong> 1.Aiming at sufficient c<strong>on</strong>diti<strong>on</strong>s, it is not difficult to show that (8) implies <strong>numerical</strong> <strong>index</strong> 1 for a Banachspace X as so<strong>on</strong> as <strong>the</strong> set ext(B X ) is large enough to determine <strong>the</strong> norm <strong>of</strong> operators <strong>on</strong> X, i.e. B X =co(ext(B X )). Actually, we may replace ext(B X ) with any subset <strong>of</strong> S X satisfying <strong>the</strong> same property. On<strong>the</strong> o<strong>the</strong>r h<str<strong>on</strong>g>and</str<strong>on</strong>g>, we may replace ext(B X ) by ext(B X ∗∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> role <strong>of</strong> ext(B X ∗) can be played by anynorming subset <strong>of</strong> S X ∗. Let us comment that this is is what we did in <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Example 2. All <strong>the</strong>seideas appear implicity in several papers (see [53, 57, 58] for example); we summarize <strong>the</strong>m in <strong>the</strong> followingpropositi<strong>on</strong>.Propositi<strong>on</strong> 6 Let X be a Banach space. Then, any <strong>of</strong> <strong>the</strong> following three c<strong>on</strong>diti<strong>on</strong>s is sufficient toensure that n(X) = 1.(a) There exists a subset C <strong>of</strong> S X such that co(C) = B X <str<strong>on</strong>g>and</str<strong>on</strong>g>for every x ∗ ∈ ext(B X ∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every c ∈ C.|x ∗ (c)| = 1(b) |x ∗∗ (x ∗ )| = 1 for every x ∗∗ ∈ ext(B X ∗∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every x ∗ ∈ ext(B X ∗).(c) There exists a norming subset A <strong>of</strong> S X ∗ such thatfor every x ∗∗ ∈ ext(B X ∗∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every a ∗ ∈ A.|x ∗∗ (a ∗ )| = 1Let us comment <strong>on</strong> <strong>the</strong> c<strong>on</strong>verse <strong>of</strong> <strong>the</strong> above result. First, c<strong>on</strong>diti<strong>on</strong> (a) is not necessary as shown by c 0 .Sec<strong>on</strong>d, it was proved in [12, Example 3.4] that c<strong>on</strong>diti<strong>on</strong> (b) is not necessary ei<strong>the</strong>r, <strong>the</strong> counterexamplebeing <strong>the</strong> space given in Example 2. Finally, we do not know if <strong>the</strong>re exists a Banach with <strong>numerical</strong> <strong>index</strong> 1in which c<strong>on</strong>diti<strong>on</strong> (c) is not satisfied.Problem 13 ([12, Remark 3.5]) Let X be a Banach space with <strong>numerical</strong> <strong>index</strong> 1. Does <strong>the</strong>re exist anorming subset A <strong>of</strong> S X ∗ such that |x ∗∗ (a ∗ )| = 1 for every x ∗∗ ∈ ext(B X ∗∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every a ∗ ∈ A?163
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. PayáNecessary c<strong>on</strong>diti<strong>on</strong>s in <strong>the</strong> spirit <strong>of</strong> McGregor’s result were given in 1999 by G. López, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g>R. Payá [53]. Their key idea was c<strong>on</strong>sidering denting points instead <strong>of</strong> general extreme points. Recall thatx 0 ∈ B X is said to be a denting point <strong>of</strong> B X if it bel<strong>on</strong>gs to slices <strong>of</strong> B X with arbitrarily small diameter.If X is a dual space <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> slices can be taken to be defined by weak ∗ -c<strong>on</strong>tinuous functi<strong>on</strong>als, <strong>the</strong>n we saythat x 0 is a weak ∗ -denting point.Propositi<strong>on</strong> 7 ([53, Lemma 1]) Let X be a Banach space with <strong>numerical</strong> <strong>index</strong> 1. Then,(a) |x ∗ (x)| = 1 for every x ∗ ∈ ext(B X ∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every denting point x ∈ B X .(b) |x ∗∗ (x ∗ )| = 1 for every x ∗∗ ∈ ext(B X ∗∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every weak ∗ -denting point x ∗ ∈ B X ∗.This result will play a key role in <strong>the</strong> next secti<strong>on</strong>.Let us comment that, like McGregor original result, <strong>the</strong> c<strong>on</strong>diti<strong>on</strong>s in Propositi<strong>on</strong> 7 are not sufficientin <strong>the</strong> infinite-dimensi<strong>on</strong>al c<strong>on</strong>text. Indeed, <strong>the</strong> space X = C([0, 1], l 2 ) does not have <strong>numerical</strong> <strong>index</strong> 1,while B X has no denting points <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong>re are no w ∗ -denting points in B X ∗. Actually, all <strong>the</strong> slices <strong>of</strong> B X<str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> w ∗ -slices <strong>of</strong> B X ∗ have diameter 2 (see [43, Lemma 2.2 <str<strong>on</strong>g>and</str<strong>on</strong>g> Example <strong>on</strong> p. 858], for instance).Anyhow, if we have a Banach space X such that B X has enough denting points (if X has <strong>the</strong> Rad<strong>on</strong>-Nikodým property, for instance), <strong>the</strong>n item (a) in <strong>the</strong> above propositi<strong>on</strong> combines with Propositi<strong>on</strong> 6 tocharacterize <strong>the</strong> <strong>numerical</strong> <strong>index</strong> 1 for X. The same is true for item (b) when B X ∗ has enough weak ∗ -denting points (if X is an Asplund space, for instance).Corollary 1 ([57, Theorem 1] <str<strong>on</strong>g>and</str<strong>on</strong>g> [58, §1]) Let X be a Banach space.(a) If X has <strong>the</strong> Rad<strong>on</strong>-Nikodým property, <strong>the</strong>n <strong>the</strong> following are equivalent:(i) X has <strong>numerical</strong> <strong>index</strong> 1.(ii) |x ∗ (x)| = 1 for every x ∗ ∈ ext(B X ∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every denting point x <strong>of</strong> B X .(iii) |x ∗∗ (x ∗ )| = 1 for every x ∗∗ ∈ ext(B X ∗∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every x ∗ ∈ ext(B X ∗).(b) If X is an Asplund space, <strong>the</strong>n <strong>the</strong> following are equivalent:(i) X has <strong>numerical</strong> <strong>index</strong> 1.(ii) |x ∗∗ (x ∗ )| = 1 for every x ∗∗ ∈ ext(B X ∗∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every weak ∗ -denting point x ∗ ∈ B X ∗.In <strong>the</strong> remaining part <strong>of</strong> this secti<strong>on</strong> we discuss o<strong>the</strong>r kind <strong>of</strong> sufficient c<strong>on</strong>diti<strong>on</strong>s for a Banach space tohave <strong>numerical</strong> <strong>index</strong> 1.The eldest <strong>of</strong> <strong>the</strong>se properties was introduced in <strong>the</strong> fifties by O. Hanner [30]: a real Banach space has<strong>the</strong> intersecti<strong>on</strong> property 3.2 (3.2.I.P. in short) if every collecti<strong>on</strong> <strong>of</strong> three mutually intersecting closed ballshas n<strong>on</strong>empty intersecti<strong>on</strong>. The 3.2.I.P. was systematically studied by J. Lindenstrauss [50] <str<strong>on</strong>g>and</str<strong>on</strong>g> Å. Lima[48], <str<strong>on</strong>g>and</str<strong>on</strong>g> typical examples <strong>of</strong> spaces with this property are L 1 (µ) <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong>ir isometric preduals. Real Banachspaces with <strong>the</strong> 3.2.I.P. have <strong>numerical</strong> <strong>index</strong> 1 since <strong>the</strong>y fulfil Propositi<strong>on</strong> 6.b (see [48, Corollary 3.3] <str<strong>on</strong>g>and</str<strong>on</strong>g>[50, Theorem 4.7]). The c<strong>on</strong>verse is false even in <strong>the</strong> finite-dimensi<strong>on</strong>al case (see [30, Remark 3.6] <str<strong>on</strong>g>and</str<strong>on</strong>g> [50,p. 47]).Ano<strong>the</strong>r isometric property, weaker than <strong>the</strong> 3.2.I.P. but still ensuring <strong>numerical</strong> <strong>index</strong> 1, was introducedby R. Fullert<strong>on</strong> in 1960 [24]. A real or complex Banach space is said to be a CL-space if its unit ball is<strong>the</strong> absolutely c<strong>on</strong>vex hull <strong>of</strong> every maximal c<strong>on</strong>vex subset <strong>of</strong> <strong>the</strong> unit sphere. If <strong>the</strong> unit ball is merely <strong>the</strong>closed absolutely c<strong>on</strong>vex hull <strong>of</strong> every maximal c<strong>on</strong>vex subset <strong>of</strong> <strong>the</strong> unit sphere, we say that <strong>the</strong> space is analmost-CL-space (J. Lindenstrauss [50] <str<strong>on</strong>g>and</str<strong>on</strong>g> Å. Lima [49]). Both definiti<strong>on</strong>s appeared <strong>on</strong>ly for real spaces,but <strong>the</strong>y extend literally to <strong>the</strong> complex case. For general informati<strong>on</strong>, we refer to <strong>the</strong> already cited papers[48, 49, 50]; more recent results can be found in [63, 71]. Let us remark that <strong>the</strong> complex space l 1 is analmost-CL-space which is not a CL-space [63, Propositi<strong>on</strong> 1], but we do not know if such an example existsin <strong>the</strong> real case.164
Numerical <strong>index</strong> <strong>of</strong> Banach spacesProblem 14 Is <strong>the</strong>re any real almost-CL-space which is not a CL-space?The fact that CL-spaces have <strong>numerical</strong> <strong>index</strong> 1 was observed by M. Acosta [1], <str<strong>on</strong>g>and</str<strong>on</strong>g> her pro<strong>of</strong> extendseasily to almost-CL-spaces (see [55, Propositi<strong>on</strong> 12]). Actually, almost-CL-spaces fulfil c<strong>on</strong>diti<strong>on</strong> (c) <strong>of</strong>Propositi<strong>on</strong> 6 as shown in [63, Lemma 3]. In <strong>the</strong> c<strong>on</strong>verse directi<strong>on</strong>, <strong>the</strong> basic examples <strong>of</strong> Banach spaceswith <strong>numerical</strong> <strong>index</strong> 1 are known to be almost-CL-spaces (see [63] <str<strong>on</strong>g>and</str<strong>on</strong>g> [10, Theorem 32.9]). Moreover,all finite-dimensi<strong>on</strong>al spaces with <strong>numerical</strong> <strong>index</strong> 1 are CL-spaces [49, Corollary 3.7], <str<strong>on</strong>g>and</str<strong>on</strong>g> a Banach spacewith <strong>the</strong> Rad<strong>on</strong>-Nikodým property <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>numerical</strong> <strong>index</strong> 1 is an almost-CL-space [57, Theorem 1]. Never<strong>the</strong>less,Banach spaces with <strong>numerical</strong> <strong>index</strong> 1 which are no almost-CL-spaces have been recently found.Actually, this happens with <strong>the</strong> space given in Example 2 [12, Example 3.4].The last c<strong>on</strong>diti<strong>on</strong> we would like to menti<strong>on</strong> is a weakening <strong>of</strong> <strong>the</strong> c<strong>on</strong>cept <strong>of</strong> almost-CL-space introducedin [12]. A Banach space X is said to be lush if for every x, y ∈ S X <str<strong>on</strong>g>and</str<strong>on</strong>g> every ε > 0, <strong>the</strong>re existsy ∗ ∈ S Y ∗ such thaty ∈ S(B X , y ∗ , ε) := {z ∈ B X : Re y ∗ (z) > 1 − ε}<str<strong>on</strong>g>and</str<strong>on</strong>g>dist ( x, co ( T S(B X , y ∗ , ε) )) < ε.In <strong>the</strong> real case, <strong>the</strong> above definiti<strong>on</strong> is equivalent to <strong>the</strong> following <strong>on</strong>e: for every x, y ∈ S X <str<strong>on</strong>g>and</str<strong>on</strong>g> everyε > 0, <strong>the</strong>re exist z ∈ S X <str<strong>on</strong>g>and</str<strong>on</strong>g> γ 1 , γ 2 ∈ R such that‖y + z‖ > 2 − ε, |γ 1 − γ 2 | = 2, ‖x + γ i z‖ 1 + ε (i = 1, 2).See Figure 2 below for an interpretati<strong>on</strong> <strong>of</strong> this property in dimensi<strong>on</strong> 2.yzx + γ 1 zxx + γ 2 zFigure 2. l 2 ∞ is lushAlmost-CL-spaces are clearly lush, <str<strong>on</strong>g>and</str<strong>on</strong>g> lush spaces have <strong>numerical</strong> <strong>index</strong> 1 [12, Propositi<strong>on</strong> 2.2]. Thec<strong>on</strong>verse <strong>of</strong> <strong>the</strong> first implicati<strong>on</strong> is not true, <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ce again <strong>the</strong> counterexample is <strong>the</strong> space X given inExample 2: as we already menti<strong>on</strong>ed, X is not an almost-CL-space, but <strong>the</strong> original pro<strong>of</strong> given in [12,Example 3.1] <strong>of</strong> <strong>the</strong> fact that n(X) = 1 passes through <strong>the</strong> lushness <strong>of</strong> <strong>the</strong> space. Actually, this example is<strong>on</strong>ly a specimen <strong>of</strong> a general family <strong>of</strong> lush subspaces <strong>of</strong> C(K) introduced in [12], namely C-rich subspaces.A subspace X <strong>of</strong> C(K) is C-rich if for every n<strong>on</strong>empty <str<strong>on</strong>g>open</str<strong>on</strong>g> subset U <strong>of</strong> K <str<strong>on</strong>g>and</str<strong>on</strong>g> every ε > 0, <strong>the</strong>re is apositive c<strong>on</strong>tinuous norm-<strong>on</strong>e functi<strong>on</strong> h with support inside U, such that <strong>the</strong> distance from h to X is lessthan ε.Theorem 4 ([12, Theorem 2.4]) Let K be a Hausdorff topological space <str<strong>on</strong>g>and</str<strong>on</strong>g> let X be a C-rich subspace<strong>of</strong> C(K). Then, X is lush <str<strong>on</strong>g>and</str<strong>on</strong>g>, <strong>the</strong>refore, n(X) = 1.165
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. PayáIn particular, this occurs for finite codimensi<strong>on</strong>al subspaces <strong>of</strong> C(K) when K has no isolated points.Actually, <strong>the</strong> following result characterizes C-rich finite-codimensi<strong>on</strong>al subspaces <strong>of</strong> an arbitrary C(K).We recall that <strong>the</strong> support <strong>of</strong> a regular measure µ ∈ C(K) ∗ issupp(µ) = ⋂ {C ⊂ K : C closed, |µ|(K \ C) = 0} .Example 5 ([12, Propositi<strong>on</strong> 2.5]) Let K be a compact Hausdorff space <str<strong>on</strong>g>and</str<strong>on</strong>g> let µ 1 , . . . , µ n ∈ C(K) ∗ .The subspacen⋂Y = ker µ iis C-rich if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if ⋃ ni=1 supp(µ i) does not intersect <strong>the</strong> set <strong>of</strong> isolated points <strong>of</strong> K.i=1We do not know if <strong>the</strong> class <strong>of</strong> lush spaces exhausts <strong>the</strong> whole class <strong>of</strong> Banach spaces with <strong>numerical</strong><strong>index</strong> 1.Problem 15 Let X be a Banach space with n(X) = 1. Is X lush?4. Renorming <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>numerical</strong> <strong>index</strong>In 2003, C. Finet, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. Payá [23] studied <strong>the</strong> <strong>numerical</strong> <strong>index</strong> from <strong>the</strong> isomorphic point <strong>of</strong>view, i.e. <strong>the</strong>y investigated <strong>the</strong> set N (X) <strong>of</strong> those values <strong>of</strong> <strong>the</strong> <strong>numerical</strong> <strong>index</strong> which can be obtainedby equivalent renormings <strong>of</strong> a Banach space X. This study has a precedent in <strong>the</strong> 1974 paper [78] byK. Tillekeratne, where it is proved that every complex space <strong>of</strong> dimensi<strong>on</strong> greater than <strong>on</strong>e can be renormedto achieve <strong>the</strong> minimum value <strong>of</strong> <strong>the</strong> <strong>numerical</strong> <strong>index</strong>; <strong>the</strong> same is true for real spaces.Propositi<strong>on</strong> 8 ([23, Propositi<strong>on</strong> 1] <str<strong>on</strong>g>and</str<strong>on</strong>g> [78, Theorem 3.1]) Let X be a Banach space <strong>of</strong> dimensi<strong>on</strong> greaterthan <strong>on</strong>e. Then 0 ∈ N (X) in <strong>the</strong> real case, e −1 ∈ N (X) in <strong>the</strong> complex case.One <strong>of</strong> <strong>the</strong> main aims <strong>of</strong> [23] is to show that N (X) is a interval for every Banach space X. To get thisresult, <strong>the</strong> authors use <strong>the</strong> c<strong>on</strong>tinuity <strong>of</strong> <strong>the</strong> mapping carrying every equivalent norm <strong>on</strong> X to its <strong>numerical</strong><strong>index</strong> with respect to a metric taken from [10, §18].Propositi<strong>on</strong> 9 ([23, Propositi<strong>on</strong> 2]) N (X) is an interval for every Banach space X.As an immediate c<strong>on</strong>sequence <strong>of</strong> <strong>the</strong> above two results, we get <strong>the</strong> following.Corollary 2 ([23, Corollary 3]) If 1 ∈ N (X) for a Banach space X <strong>of</strong> dimensi<strong>on</strong> greater than <strong>on</strong>e, <strong>the</strong>nN (X) = [0, 1] in <strong>the</strong> real case <str<strong>on</strong>g>and</str<strong>on</strong>g> N (X) = [e −1 , 1] in <strong>the</strong> complex case.Since n(l m ∞) = 1 for every m, <strong>the</strong> following particular case arises.Corollary 3 ([78, Theorem 3.2]) Let m be an integer larger than 1. ThenN (R m ) = [0, 1] <str<strong>on</strong>g>and</str<strong>on</strong>g> N (C m ) = [e −1 , 1].Now, <strong>on</strong>e may ask if <strong>the</strong> above result is also true in <strong>the</strong> infinite-dimensi<strong>on</strong>al c<strong>on</strong>text, equivalently,whe<strong>the</strong>r or not every Banach space can be equivalently renormed to have <strong>numerical</strong> <strong>index</strong> 1. The answer isnegative, as shown in <strong>the</strong> already cited paper [53].Theorem 5 ([53, Theorem 3]) Let X be an infinite-dimensi<strong>on</strong>al real Banach space with 1 ∈ N (X). If Xhas <strong>the</strong> Rad<strong>on</strong>-Nikodým property, <strong>the</strong>n X c<strong>on</strong>tains l 1 . If X is an Asplund space, <strong>the</strong>n X ∗ c<strong>on</strong>tains l 1 .166
Numerical <strong>index</strong> <strong>of</strong> Banach spacesIt follows that infinite-dimensi<strong>on</strong>al real reflexive spaces cannot be renormed to have <strong>numerical</strong> <strong>index</strong> 1.But even more is true.Corollary 4 ([53, Corollary 5]) Let X be an infinite-dimensi<strong>on</strong>al real Banach space. If X ∗∗ /X is separable,<strong>the</strong>n 1 /∈ N (X).It is easy to explain how Theorem 5 was proved in [53]. Namely, <strong>the</strong> authors used Propositi<strong>on</strong> 7, <strong>the</strong>well-known facts that <strong>the</strong> unit ball <strong>of</strong> a space with <strong>the</strong> Rad<strong>on</strong>-Nikodým property has many denting points<str<strong>on</strong>g>and</str<strong>on</strong>g> that <strong>the</strong> dual unit ball <strong>of</strong> an Asplund space has many weak ∗ -denting points (see [11], for instance), <str<strong>on</strong>g>and</str<strong>on</strong>g><strong>the</strong> following sufficient c<strong>on</strong>diti<strong>on</strong> for a real Banach space to c<strong>on</strong>tain ei<strong>the</strong>r c 0 or l 1 .Lemma 1 ([53, Propositi<strong>on</strong> 2]) Let X be a real Banach space, <str<strong>on</strong>g>and</str<strong>on</strong>g> assume that <strong>the</strong>re is an infinite setA ⊂ S X such that |x ∗ (a)| = 1 for every a ∈ A <str<strong>on</strong>g>and</str<strong>on</strong>g> every x ∗ ∈ ext(B X ∗). Then X c<strong>on</strong>tains c 0 or l 1 .Thus, <strong>the</strong> first <str<strong>on</strong>g>open</str<strong>on</strong>g> questi<strong>on</strong> in this line is <strong>the</strong> following.Problem 16 Characterize those Banach spaces which can be equivalently renormed to have <strong>numerical</strong><strong>index</strong> 1.We also propose to study separately necessary <str<strong>on</strong>g>and</str<strong>on</strong>g> sufficient c<strong>on</strong>diti<strong>on</strong>s for a Banach space to berenormable with <strong>numerical</strong> <strong>index</strong> 1. With respect to necessary c<strong>on</strong>diti<strong>on</strong>s, we have obtained two in <strong>the</strong>real case, namely Theorem 5 <str<strong>on</strong>g>and</str<strong>on</strong>g> Corollary 4. It is not known if <strong>the</strong>y are valid in <strong>the</strong> complex case; actually,<strong>the</strong> following especial case remains <str<strong>on</strong>g>open</str<strong>on</strong>g>.Problem 17 Does <strong>the</strong>re exist an infinite-dimensi<strong>on</strong>al complex reflexive space which can be renormed tohave <strong>numerical</strong> <strong>index</strong> 1?For more necessary c<strong>on</strong>diti<strong>on</strong>s, we suggest to study <strong>the</strong> following two <str<strong>on</strong>g>questi<strong>on</strong>s</str<strong>on</strong>g>. The first <strong>on</strong>e has apositive answer for real almost-CL-spaces [63, Theorem 5].Problem 18 Let X be an infinite-dimensi<strong>on</strong>al (real) Banach space satisfying that 1 ∈ N (X). Does X ∗c<strong>on</strong>tain l 1 ?Problem 19 Let X be an infinite-dimensi<strong>on</strong>al (real) Banach space satisfying that 1 ∈ N (X). Does Xc<strong>on</strong>tain c 0 or l 1 ?Let us remark that we do not know <strong>of</strong> any n<strong>on</strong>-trivial sufficient c<strong>on</strong>diti<strong>on</strong> for a Banach space to berenormable with <strong>numerical</strong> <strong>index</strong> 1. We propose <strong>the</strong> following <strong>on</strong>es to be checked.Problem 20 Let X be a Banach space c<strong>on</strong>taining an infinite-dimensi<strong>on</strong>al subspace Y with 1 ∈ N (Y ). Isit true that 1 ∈ N (X)?One may c<strong>on</strong>sider many especial cases.Problem 21 Let X be a Banach space c<strong>on</strong>taining a subspace isomorphic to ei<strong>the</strong>r c 0 , l 1 , C[0, 1], orL 1 [0, 1]. Is it true that 1 ∈ N (X)?The following questi<strong>on</strong> is especially interesting since in view <strong>of</strong> Problem 18 it might lead to a characterizati<strong>on</strong><strong>of</strong> Banach spaces that can be renormed to have <strong>numerical</strong> <strong>index</strong> 1.Problem 22 Let X be a Banach space such that l 1 ⊆ X ∗ . Is is true that 1 ∈ N (X)?We finish this secti<strong>on</strong> by showing that <strong>the</strong> value 1 <strong>of</strong> <strong>the</strong> <strong>numerical</strong> <strong>index</strong> is very particular. Indeed, it isproved in [23] that “most” Banach spaces can be renormed to achieve any possible value for <strong>the</strong> <strong>numerical</strong><strong>index</strong> except eventually 1. Recall that a system {(x λ , x ∗ λ )} λ∈Λ ⊂ X × X ∗ is said to be biorthog<strong>on</strong>al ifx ∗ λ (x µ) = δ λ,µ for λ, µ ∈ Λ, <str<strong>on</strong>g>and</str<strong>on</strong>g> l<strong>on</strong>g if <strong>the</strong> cardinality <strong>of</strong> Λ coincides with <strong>the</strong> density character <strong>of</strong> X.167
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. PayáTheorem 6 ([23, Theorem 10]) Let X be a Banach space admitting a l<strong>on</strong>g biorthog<strong>on</strong>al system. Thensup N (X) = 1. Therefore, when <strong>the</strong> dimensi<strong>on</strong> <strong>of</strong> X is greater than <strong>on</strong>e, N (X) ⊃ [0, 1[ in <strong>the</strong> real case<str<strong>on</strong>g>and</str<strong>on</strong>g> N (X) ⊃ [e −1 , 1[ in <strong>the</strong> complex case.Typical examples <strong>of</strong> Banach spaces admitting a l<strong>on</strong>g biorthog<strong>on</strong>al system are WCG spaces (see [18]).For instance, if X ∗∗ /X is separable, <strong>the</strong>n <strong>the</strong> Banach space X is WCG (see [80, Theorem 3], for example)while, in <strong>the</strong> real case, 1 /∈ N (X) unless X is finite-dimensi<strong>on</strong>al (see Corollary 4). Therefore, in manycases <strong>on</strong>e <strong>of</strong> <strong>the</strong> inclusi<strong>on</strong>s <strong>of</strong> Theorem 6 becomes an equality.Corollary 5 ([23, Corollary 11]) Let X be an infinite-dimensi<strong>on</strong>al real Banach space such that X ∗∗ /X isseparable. Then N (X) = [0, 1[.Let us comment that Theorem 6 is proved by using a geometrical property that was introduced byJ. Lindenstrauss in <strong>the</strong> study <strong>of</strong> norn-attaining operators [51] <str<strong>on</strong>g>and</str<strong>on</strong>g> called property α by W. Schachermayer[75]. It is known that, under <strong>the</strong> c<strong>on</strong>tinuum hypo<strong>the</strong>sis, <strong>the</strong>re are Banach spaces which cannot be renormedwith property α [27, 67]. Never<strong>the</strong>less, B. Godun <str<strong>on</strong>g>and</str<strong>on</strong>g> S. Troyanski proved in [27, Theorem 1] that thisrenorming is possible for Banach spaces admitting a l<strong>on</strong>g biorthog<strong>on</strong>al system; as far as we know, this is<strong>the</strong> largest class <strong>of</strong> spaces for which renorming with property α is possible.The questi<strong>on</strong> arises if <strong>the</strong> assumpti<strong>on</strong> <strong>of</strong> having a l<strong>on</strong>g biorthog<strong>on</strong>al system in Theorem 6 can bedropped.Problem 23 Is is true that sup N (X) = 1 for every Banach space X?It is also studied in [23] <strong>the</strong> relati<strong>on</strong>ship between <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> so-called property β[51, 75]. C<strong>on</strong>trary to property α, property β is isomorphically trivial (J. Partingt<strong>on</strong> [69]), but it does notproduce such a good result as Theorem 6. At least, it can be used to prove that N (X) does not reduces to apoint when <strong>the</strong> dimensi<strong>on</strong> <strong>of</strong> X is greater than <strong>on</strong>e.Theorem 7 ([23, Theorem 9]) Let X be a Banach space with dim(X) > 1. Then N (X) ⊃ [0, 1/3[ in <strong>the</strong>real case <str<strong>on</strong>g>and</str<strong>on</strong>g> N (X) ⊃ [e −1 , 1/2[ in <strong>the</strong> complex case.5. Real Banach spaces with <strong>numerical</strong> <strong>index</strong> zeroAs we commented in <strong>the</strong> introducti<strong>on</strong>, real Banach spaces underlying complex Banach spaces as well asreal Hilbert spaces <strong>of</strong> dimensi<strong>on</strong> greater than <strong>on</strong>e, have <strong>numerical</strong> <strong>index</strong> 0. By Propositi<strong>on</strong> 3, <strong>the</strong> absolutesum <strong>of</strong> such a space <str<strong>on</strong>g>and</str<strong>on</strong>g> any o<strong>the</strong>r real Banach space has <strong>numerical</strong> <strong>index</strong> 0.The general <str<strong>on</strong>g>open</str<strong>on</strong>g> questi<strong>on</strong> in this secti<strong>on</strong> is <strong>the</strong> following.Problem 24 Find characterizati<strong>on</strong>s <strong>of</strong> Banach spaces with <strong>numerical</strong> <strong>index</strong> 0 which do not involve operators.A sufficient c<strong>on</strong>diti<strong>on</strong> which generalizes all <strong>the</strong> introductory examples is <strong>the</strong> following easy result <strong>of</strong>M. Martín, J. Merí <str<strong>on</strong>g>and</str<strong>on</strong>g> A. Rodríguez Palacios [60]. We say that a real vector space has a complex structureif it is <strong>the</strong> real space underlying a complex vector space.Propositi<strong>on</strong> 10 ([60, Propositi<strong>on</strong> 2.1]) Let X be a real Banach space, <str<strong>on</strong>g>and</str<strong>on</strong>g> let Y, Z be closed subspaces<strong>of</strong> X, with Z ≠ 0. Suppose that Z is endowed with a complex structure, that X = Y ⊕ Z, <str<strong>on</strong>g>and</str<strong>on</strong>g> that <strong>the</strong>equality ∥ ∥y + e iρ z ∥ ∥ = ‖y + z‖ holds for every (ρ, y, z) ∈ R × Y × Z. Then n(X) = 0.It looks like if a necessary c<strong>on</strong>diti<strong>on</strong> for <strong>numerical</strong> <strong>index</strong> 0 could be <strong>the</strong> emergence <strong>of</strong> a subspace withsome kind <strong>of</strong> complex structure. As a matter <strong>of</strong> fact, <strong>the</strong> following example shows that this is not <strong>the</strong> case.168
Numerical <strong>index</strong> <strong>of</strong> Banach spacesExample 6 ([60, Example 2.2]) There exists a real Banach space X with <strong>numerical</strong> <strong>index</strong> 0 which is polyhedral,i.e. <strong>the</strong> intersecti<strong>on</strong> <strong>of</strong> B X with any finite-dimensi<strong>on</strong>al subspace <strong>of</strong> X is <strong>the</strong> c<strong>on</strong>vex hull <strong>of</strong> a finiteset <strong>of</strong> points. Therefore, X does not c<strong>on</strong>tain any isometric copy <strong>of</strong> C.The above example has <strong>the</strong> additi<strong>on</strong>al interest that <strong>the</strong> <strong>numerical</strong> radius is a norm <strong>on</strong> L(X), i.e. <strong>the</strong> <strong>on</strong>lyoperator with <strong>numerical</strong> radius 0 is <strong>the</strong> zero operator. It could be <strong>the</strong> case that a Banach space in which<strong>the</strong>re is a n<strong>on</strong>-null operator with <strong>numerical</strong> radius 0 has a subspace with some kind <strong>of</strong> complex structure.We recall that a bounded linear operator T <strong>on</strong> a (real or complex) Banach space X is skew-hermitian ifRe V (T ) = {0}; we write Z(X) for <strong>the</strong> (possibly null) closed subspace <strong>of</strong> L(X) c<strong>on</strong>sisting <strong>of</strong> skewhermitianoperators <strong>on</strong> X. In <strong>the</strong> real case, T is skew-hermitian if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if v(T ) = 0; when <strong>the</strong> spaceX is complex, an operator T is hermitian if V (T ) ⊂ R, i.e. <strong>the</strong> operator i T is skew-hermitian. Hermitianoperators have been deeply studied since <strong>the</strong> sixties <str<strong>on</strong>g>and</str<strong>on</strong>g> many results <strong>on</strong> Banach algebras depend <strong>on</strong> <strong>the</strong>m;we refer to [9, 10] for more informati<strong>on</strong>. Also, skew-hermitian operators have been widely discussed in<strong>the</strong> seventies <str<strong>on</strong>g>and</str<strong>on</strong>g> eighties, especially in <strong>the</strong> finite-dimensi<strong>on</strong>al case; more informati<strong>on</strong> can be found in <strong>the</strong>papers by H. Rosenthal [73, 74] <str<strong>on</strong>g>and</str<strong>on</strong>g> references <strong>the</strong>rein.Problem 25 Let X be a real space which has a n<strong>on</strong>-null skew-hermitian operator. Does X c<strong>on</strong>tain asubspace with a complex structure?Let us give a clarifying example. If H is a n-dimensi<strong>on</strong>al Hilbert space, it is easy to check that Z(H)is <strong>the</strong> space <strong>of</strong> skew-symmetric operators <strong>on</strong> H (i.e. T ∗ = −T in <strong>the</strong> Hilbert space sense), so it identifieswith <strong>the</strong> space <strong>of</strong> skew-symmetric matrixes A(n). It is a classical result from <strong>the</strong> <strong>the</strong>ory <strong>of</strong> linear algebrathat a n × n matrix A bel<strong>on</strong>gs to A(n) if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if exp(ρA) is an orthog<strong>on</strong>al matrix for every ρ ∈ R (see[3, Corollary 8.5.10] for instance).It is shown in [9, §3] that <strong>the</strong> above fact extends to general Banach spaces. Indeed, for an arbitraryBanach space X <str<strong>on</strong>g>and</str<strong>on</strong>g> an operator T ∈ L(X), by making use <strong>of</strong> <strong>the</strong> “exp<strong>on</strong>ential formula”sup Re V (T ) = supα>0it is easy to prove that <strong>the</strong> following are equivalent:(i) T is skew-hermitian,(ii) exp(ρT ) is an <strong>on</strong>to isometry for every ρ ∈ R.log ‖ exp(αT )‖αFrom now <strong>on</strong>, we will restrict ourselves to finite-dimensi<strong>on</strong>al spaces. Here, <strong>the</strong> above result reminds<strong>the</strong> <strong>the</strong>ory <strong>of</strong> Lie groups <str<strong>on</strong>g>and</str<strong>on</strong>g> Lie algebras, for which <strong>the</strong> group <strong>of</strong> orthog<strong>on</strong>al matrices <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> space <strong>of</strong>skew-symmetric matrices are distinguished examples. Actually, <strong>the</strong> group <strong>of</strong> all <strong>the</strong> isometries <strong>on</strong> a finitedimensi<strong>on</strong>alreal space X is a Lie group whose associated Lie algebra is Z(X) [73, Theorem 1.4 <str<strong>on</strong>g>and</str<strong>on</strong>g>Propositi<strong>on</strong> 1.5]. With this in mind <str<strong>on</strong>g>and</str<strong>on</strong>g> using results from <strong>the</strong> <strong>the</strong>ory <strong>of</strong> Lie groups, it is proved in [73,Theorem 3.8] that <strong>the</strong> following are equivalent for a finite-dimensi<strong>on</strong>al real space X:(i) <strong>the</strong> <strong>numerical</strong> <strong>index</strong> <strong>of</strong> X is 0,(ii) <strong>the</strong>re are infinitely many isometries <strong>on</strong> X.The main <str<strong>on</strong>g>open</str<strong>on</strong>g> questi<strong>on</strong> in this secti<strong>on</strong> is <strong>the</strong> following.Problem 26 Describe <strong>the</strong> finite-dimensi<strong>on</strong>al real Banach spaces with <strong>numerical</strong> <strong>index</strong> 0.The next result follows this line. In particular, it shows that finite-dimensi<strong>on</strong>al normed spaces with<strong>numerical</strong> <strong>index</strong> 0 wear some kind <strong>of</strong> complex structure.Theorem 8 ([73, Corollary 3.7] <str<strong>on</strong>g>and</str<strong>on</strong>g> [60, Theorem 2.4]) Let X be a finite-dimensi<strong>on</strong>al real Banach space.Then, <strong>the</strong> following are equivalent:,169
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. Payá(i) The <strong>numerical</strong> <strong>index</strong> <strong>of</strong> X is zero.(ii) There are n<strong>on</strong>zero complex vector spaces X 1 , . . . , X m , a real vector space X 0 , <str<strong>on</strong>g>and</str<strong>on</strong>g> positive integernumbers q 1 , . . . , q m such that X = X 0 ⊕ X 1 ⊕ · · · ⊕ X m <str<strong>on</strong>g>and</str<strong>on</strong>g>∥∥x 0 + e iq 1ρ x 1 + · · · + e iq mρ x m∥ ∥ = ‖x 0 + x 1 + · · · + x m ‖for all ρ ∈ R, x j ∈ X j (j = 0, 1, . . . , m).Some remarks are pertinent. First, <strong>the</strong> above result shows that a finite-dimensi<strong>on</strong>al space with <strong>numerical</strong><strong>index</strong> 0 c<strong>on</strong>tains complex subspaces (at least <strong>on</strong>e), whose complex structures are well related <strong>on</strong>e to <strong>the</strong>o<strong>the</strong>rs <str<strong>on</strong>g>and</str<strong>on</strong>g> to <strong>the</strong> n<strong>on</strong>-complex part. Sec<strong>on</strong>d, <strong>the</strong> above result is taken literally from <strong>the</strong> paper [60], but<strong>the</strong> same equivalence with arbitrary real numbers q 1 , . . . , q m had appeared in <strong>the</strong> 1985 Rosenthal’s paper[73]; <strong>the</strong> fact that <strong>the</strong> q j ’s can be taken integers is new from [60] <str<strong>on</strong>g>and</str<strong>on</strong>g> it uses <strong>the</strong> classical Kr<strong>on</strong>ecker’sApproximati<strong>on</strong> Theorem (see [32, Theorem 442] for instance). Third, it can be asked if <strong>the</strong> number <strong>of</strong>complex spaces in <strong>the</strong> <strong>the</strong>orem can be reduced to <strong>on</strong>e, <str<strong>on</strong>g>and</str<strong>on</strong>g> so Propositi<strong>on</strong> 10 would be an equivalence in <strong>the</strong>finite-dimensi<strong>on</strong>al case. The answer is negative.Example 7 ([60, Example 2.8]) The space R 4 with norm‖(a, b, c, d)‖ = 1 4∫ 2π0∣∣Re ( e 2it (a + ib) + e it (c + id) )∣ ∣ dt (a, b, c, d ∈ R)has <strong>numerical</strong> <strong>index</strong> 0, but <strong>the</strong> number <strong>of</strong> complex spaces in Theorem 8.ii cannot be reduced to <strong>on</strong>e.Such an example as <strong>the</strong> above is not possible in dimensi<strong>on</strong>s two or three. Actually, in this case, Theorem8 takes a more suitable form [73, Theorem 3.1]. Let X be a real Banach space with <strong>numerical</strong> <strong>index</strong> 0.(a) If dim(X) = 2, <strong>the</strong>n X is isometrically isomorphic to <strong>the</strong> two-dimensi<strong>on</strong>al real Hilbert space.(b) If dim(X) = 3, <strong>the</strong>n X is an absolute sum <strong>of</strong> R <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> two-dimensi<strong>on</strong>al real Hilbert space.Our next aim is to discuss some <str<strong>on</strong>g>questi<strong>on</strong>s</str<strong>on</strong>g> related to <strong>the</strong> Lie algebra <strong>of</strong> skew-hermitian operators Z(X)<strong>of</strong> an arbitrary n-dimensi<strong>on</strong>al space which. The main related <str<strong>on</strong>g>open</str<strong>on</strong>g> questi<strong>on</strong> is <strong>the</strong> following.Problem 27 Figure out what are <strong>the</strong> possible values for <strong>the</strong> dimensi<strong>on</strong> <strong>of</strong> Z(X) when dim(X) = n.Let us fix a n-dimensi<strong>on</strong>al real Banach space X. It follows from a <strong>the</strong>orem <strong>of</strong> Auerbach [72, Theorem9.5.1], that <strong>the</strong>re exists an inner product (·|·) <strong>on</strong> X such that every skew-hermitian operator <strong>on</strong> Xremains skew-hermitian (hence skew-symmetric) <strong>on</strong> H := (X, (·|·)). Then, by just fixing an orth<strong>on</strong>ormalbasis <strong>of</strong> H, we get an identificati<strong>on</strong> <strong>of</strong> Z(X) with a Lie subalgebra <strong>of</strong> <strong>the</strong> Lie algebra A(n). Therefore,dim ( Z(X) ) n(n − 1) .2The equality holds if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if X is a Hilbert space (see [73, Theorem 3.2] or [60, Corollary 2.7]). It is agood questi<strong>on</strong> whe<strong>the</strong>r or not all <strong>the</strong> intermediate numbers are possible values for <strong>the</strong> dimensi<strong>on</strong> <strong>of</strong> Z(X).The answer is negative, as a c<strong>on</strong>sequence <strong>of</strong> Theorem 3.2 in Rosenthal’s paper [73], which reads as follows.(a) If dim ( Z(X) ) > (n−1)(n−2)2, <strong>the</strong>n X is a Hilbert space <str<strong>on</strong>g>and</str<strong>on</strong>g>, <strong>the</strong>refore, dim ( Z(X) ) = n(n−1)2.(b) dim ( Z(X) ) = (n−1)(n−2)2if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if X is a n<strong>on</strong>-Euclidean absolute sum <strong>of</strong> R <str<strong>on</strong>g>and</str<strong>on</strong>g> a Hilbertspace <strong>of</strong> dimensi<strong>on</strong> n − 1.For low dimensi<strong>on</strong>s, Problem 27 has been solved in [74]. When <strong>the</strong> dimensi<strong>on</strong> <strong>of</strong> X is 3, <strong>the</strong> above resultleaves <strong>on</strong>ly <strong>the</strong> following possible values for <strong>the</strong> dimensi<strong>on</strong> <strong>of</strong> Z(X): 0 as for X = l 3 ∞, 1 as for R ⊕ 1 C,<str<strong>on</strong>g>and</str<strong>on</strong>g> 3 as for l 3 2. When <strong>the</strong> dimensi<strong>on</strong> <strong>of</strong> X is 4, <strong>the</strong> possible values <strong>of</strong> <strong>the</strong> dimensi<strong>on</strong> <strong>of</strong> Z(X) allowed by <strong>the</strong>above result are 0, 1, 2, 3, 6; all <strong>of</strong> <strong>the</strong>m are possible [74, pp. 443]. The first dimensi<strong>on</strong> in which Problem 27is <str<strong>on</strong>g>open</str<strong>on</strong>g> is n = 5.Problem 28 What are <strong>the</strong> possible values for <strong>the</strong> dimensi<strong>on</strong> <strong>of</strong> Z(X) when X is a 5-dimensi<strong>on</strong>al realBanach space?170
Numerical <strong>index</strong> <strong>of</strong> Banach spaces6. Asymptotic behavior <strong>of</strong> <strong>the</strong> set <strong>of</strong> finite-dimensi<strong>on</strong>al spaceswith <strong>numerical</strong> <strong>index</strong> <strong>on</strong>eLet us start <strong>the</strong> secti<strong>on</strong> by recalling that most <strong>of</strong> <strong>the</strong> sufficient <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> necessary c<strong>on</strong>diti<strong>on</strong>s for having<strong>numerical</strong> <strong>index</strong> 1 given in secti<strong>on</strong> 3 are actually characterizati<strong>on</strong>s in <strong>the</strong> finite-dimensi<strong>on</strong>al case. For <strong>the</strong>c<strong>on</strong>venience <strong>of</strong> <strong>the</strong> reader, we summarize <strong>the</strong>m in <strong>the</strong> following paragraph.Let X be a finite-dimensi<strong>on</strong>al real normed space. Then, <strong>the</strong> following are equivalent.(i) X has <strong>numerical</strong> <strong>index</strong> 1.(ii) |x ∗ (x)| = 1 for all <strong>the</strong> extreme points x ∗ <strong>of</strong> B X ∗ <str<strong>on</strong>g>and</str<strong>on</strong>g> x <strong>of</strong> B X .(iii) X is a CL-space, i.e. B X = co(F ∪ −F ) for every maximal c<strong>on</strong>vex subset F<strong>of</strong> S X .(iv) X is lush, i.e. for every x, y ∈ S X , <strong>the</strong>re are z ∈ S X <str<strong>on</strong>g>and</str<strong>on</strong>g> γ 1 , γ 2 ∈ R such that‖y + z‖ = 2, |γ 1 − γ 2 | = 2, <str<strong>on</strong>g>and</str<strong>on</strong>g> ‖x + γ i z‖ 1 (i = 1, 2).Our aim in this secti<strong>on</strong> is to c<strong>on</strong>sider <strong>the</strong> asymptotic behavior (as <strong>the</strong> dimensi<strong>on</strong> grows to infinity)<strong>of</strong> some parameters related to <strong>the</strong> Banach-Mazur distance for <strong>the</strong> family <strong>of</strong> finite-dimensi<strong>on</strong>al real normedspaces with <strong>numerical</strong> <strong>index</strong> 1. Let us write N m for <strong>the</strong> space <strong>of</strong> all m-dimensi<strong>on</strong>al normed spaces endowedwith <strong>the</strong> Banach-Mazur distanced(X, Y ) = inf{‖T ‖ ‖T −1 ‖ : T : X −→ Y isomorphism} (X, Y ∈ N m ),<str<strong>on</strong>g>and</str<strong>on</strong>g> let us write M m for <strong>the</strong> subset c<strong>on</strong>sisting <strong>of</strong> those m-dimensi<strong>on</strong>al spaces with <strong>numerical</strong> <strong>index</strong> 1. Ouraim is to study some <str<strong>on</strong>g>questi<strong>on</strong>s</str<strong>on</strong>g> related to <strong>the</strong>se two spaces. As far as we know, <strong>the</strong> first result <strong>of</strong> this kindwas given very recently by T. Oikhberg [68].Theorem 9 ([68, Theorem 4.1]) There exists a universal positive c<strong>on</strong>stant c such thatfor every m 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> every X ∈ M m .d(X, l m 2 ) c m 1 4It is well-known that d(l m 1 , l m 2 ) = d(l m ∞, l m 2 ) = √ m for every m > 1 (see [25, pp. 720] for instance).Therefore, <strong>the</strong> following questi<strong>on</strong> arises naturally.Problem 29 Does <strong>the</strong>re exists a universal c<strong>on</strong>stant c > 0 such thatfor every m 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> every X ∈ M m ?d(X, l m 2 ) c √ mIt was observed in [68, pp. 622] that <strong>the</strong> answer to this questi<strong>on</strong> is positive for <strong>the</strong> spaces with <strong>the</strong>3.2.I.P., even with c = 1. This is a c<strong>on</strong>sequence <strong>of</strong> <strong>the</strong> 1981 result by A. Hansen <str<strong>on</strong>g>and</str<strong>on</strong>g> Å. Lima [31] that<strong>the</strong>se spaces are c<strong>on</strong>structed starting from <strong>the</strong> real line <str<strong>on</strong>g>and</str<strong>on</strong>g> producing successively l ∞ <str<strong>on</strong>g>and</str<strong>on</strong>g>/or l 1 sums. But,as we already menti<strong>on</strong>ed, not every element <strong>of</strong> M m has <strong>the</strong> 3.2.I.P.Finally, we would like to propose some related <str<strong>on</strong>g>questi<strong>on</strong>s</str<strong>on</strong>g>.Problem 30 What is <strong>the</strong> diameter <strong>of</strong> M m ? Is it (asymptotically) close to <strong>the</strong> diameter <strong>of</strong> N m ?Problem 31 What is <strong>the</strong> biggest possible distance from an element <strong>of</strong> N m to <strong>the</strong> set M m ?171
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. Payá7. Relati<strong>on</strong>ship to <strong>the</strong> Daugavet property.In every Banach space with <strong>the</strong> Rad<strong>on</strong>-Nikodým property (in particular in every reflexive space) <strong>the</strong> unitball must have denting points. There are Banach spaces X (as C[0, 1], L 1 [0, 1], <str<strong>on</strong>g>and</str<strong>on</strong>g> many o<strong>the</strong>rs) with anextremely opposite property: for every x ∈ S X <str<strong>on</strong>g>and</str<strong>on</strong>g> for arbitrarily small ε > 0, <strong>the</strong> closure <strong>of</strong>co ( B X \ (x + (2 − ε)B X ) )equals to <strong>the</strong> whole B X (see Figure 3 below). This geometric property <strong>of</strong> <strong>the</strong> space is equivalent to <strong>the</strong>xB X \ ( x + (2 − ε)B X)Figure 3. l 2 ∞ does not have <strong>the</strong> Daugavet propertyfollowing exotic property <strong>of</strong> operators <strong>on</strong> X: for every compact operator T : X −→ X, <strong>the</strong> so-calledDaugavet equati<strong>on</strong>‖Id + T ‖ = 1 + ‖T ‖(DE)holds. This property <strong>of</strong> C[0, 1] was discovered by I. Daugavet in 1963 <str<strong>on</strong>g>and</str<strong>on</strong>g> is called <strong>the</strong> Daugavet property[42, 43]. Over <strong>the</strong> years, <strong>the</strong> validity <strong>of</strong> <strong>the</strong> Daugavet equati<strong>on</strong> was proved for some classes <strong>of</strong> operators <strong>on</strong>various spaces, including weakly compact operators <strong>on</strong> C(K) <str<strong>on</strong>g>and</str<strong>on</strong>g> L 1 (µ) provided that K is perfect <str<strong>on</strong>g>and</str<strong>on</strong>g> µdoes not have any atoms (see [81] for an elementary approach), <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong> certain functi<strong>on</strong> algebras such as<strong>the</strong> disk algebra A(D) or <strong>the</strong> algebra <strong>of</strong> bounded analytic functi<strong>on</strong>s H ∞ [82, 84]. In <strong>the</strong> nineties, new ideaswere infused into this field <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> geometry <strong>of</strong> Banach spaces having <strong>the</strong> Daugavet property was studied;we cite <strong>the</strong> papers <strong>of</strong> V. Kadets, R. Shvidkoy, G. Sirotkin, <str<strong>on</strong>g>and</str<strong>on</strong>g> D. Werner [43] <str<strong>on</strong>g>and</str<strong>on</strong>g> R. Shvidkoy [77] asrepresentatives. Let us comment that <strong>the</strong> original definiti<strong>on</strong> <strong>of</strong> Daugavet property given in [42, 43] <strong>on</strong>lyrequired rank-<strong>on</strong>e operators to satisfy (DE) <str<strong>on</strong>g>and</str<strong>on</strong>g>, in such a case, this equati<strong>on</strong> also holds for every boundedoperator which does not fix a copy <strong>of</strong> l 1 [77].Although <strong>the</strong> Daugavet property is <strong>of</strong> isometric nature, it induces various isomorphic restricti<strong>on</strong>s. Forinstance, a Banach space with <strong>the</strong> Daugavet property c<strong>on</strong>tains l 1 [43], it does not have unc<strong>on</strong>diti<strong>on</strong>al basis(V. Kadets [38]) <str<strong>on</strong>g>and</str<strong>on</strong>g>, moreover, it does not isomorphically embed into an unc<strong>on</strong>diti<strong>on</strong>al sum <strong>of</strong> Banachspaces without a copy <strong>of</strong> l 1 [77]. It is worthwhile to remark that <strong>the</strong> latest result c<strong>on</strong>tinues a line <strong>of</strong>generalizati<strong>on</strong> ([39], [41], [43]) <strong>of</strong> <strong>the</strong> well known <strong>the</strong>orem by A. Pełczyński [70] that L 1 [0, 1] (<str<strong>on</strong>g>and</str<strong>on</strong>g> soC[0, 1]) does not embed into a space with unc<strong>on</strong>diti<strong>on</strong>al basis.172
Numerical <strong>index</strong> <strong>of</strong> Banach spacesThe state-<strong>of</strong>-<strong>the</strong>-art <strong>on</strong> <strong>the</strong> Daugavet property can be found in [83]; for very recent results we refer <strong>the</strong>reader to [5, 6, 37, 40, 44] <str<strong>on</strong>g>and</str<strong>on</strong>g> references <strong>the</strong>rein.Let us explain <strong>the</strong> relati<strong>on</strong> between (DE) <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> <strong>numerical</strong> range <strong>of</strong> an operator. In <strong>the</strong> aforementi<strong>on</strong>edpaper [20] by J. Duncan, C. McGregor, J. Pryce, <str<strong>on</strong>g>and</str<strong>on</strong>g> A. White, it was deduced from formula (3) <strong>on</strong> page 160that an operator T <strong>on</strong> a Banach space X satisfies (DE) if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if sup Re V (T ) = ‖T ‖. Therefore,v(T ) = ‖T ‖ if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly <strong>the</strong> following equality holdsmax ‖Id + ω T ‖ = 1 + ‖T ‖(aDE)ω∈T(see [61, Lemma 2.3] for an explicit pro<strong>of</strong>). Therefore, it was known since 1970 that every bounded linearoperator <strong>on</strong> C(K) or L 1 (µ) satisfies (aDE), a fact that was rediscovered <str<strong>on</strong>g>and</str<strong>on</strong>g> reproved in some papers from<strong>the</strong> eighties <str<strong>on</strong>g>and</str<strong>on</strong>g> nineties as <strong>the</strong> <strong>on</strong>es by Y. Abramovich [2], J. Holub [34], <str<strong>on</strong>g>and</str<strong>on</strong>g> K. Schmidt [76].This latest equati<strong>on</strong> was named as <strong>the</strong> alternative Daugavet equati<strong>on</strong> by M. Martín <str<strong>on</strong>g>and</str<strong>on</strong>g> T. Oikhbergin [61], where <strong>the</strong> following property was introduced. A Banach space X is said to have <strong>the</strong> alternativeDaugavet property if every rank-<strong>on</strong>e operator <strong>on</strong> X satisfies (aDE). In such a case, every weakly compactoperator <strong>on</strong> X also satisfies (aDE) [61, Theorem 2.2]. Therefore, X has <strong>the</strong> alternative Daugavet propertyif <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if v(T ) = ‖T ‖ for every weakly compact operator T <strong>on</strong> X.Let us comment that, c<strong>on</strong>trary to <strong>the</strong> Daugavet property, this property depends up<strong>on</strong> <strong>the</strong> base field(e.g. C has it as a complex space but not as a real space). For more informati<strong>on</strong> <strong>on</strong> <strong>the</strong> alternative Daugavetproperty we refer to <strong>the</strong> already cited paper [61] <str<strong>on</strong>g>and</str<strong>on</strong>g> also to [58]. From <strong>the</strong> former <strong>on</strong>e we take <strong>the</strong> followinggeometric characterizati<strong>on</strong>s <strong>of</strong> <strong>the</strong> alternative Daugavet property.Propositi<strong>on</strong> 11 ([61, Propositi<strong>on</strong>s 2.1 <str<strong>on</strong>g>and</str<strong>on</strong>g> 2.6]) Let X be a Banach space. Then, <strong>the</strong> following are equivalent.(i) X has <strong>the</strong> alternative Daugavet property.(ii) For all x 0 ∈ S X , x ∗ 0 ∈ S X ∗ <str<strong>on</strong>g>and</str<strong>on</strong>g> ε > 0, <strong>the</strong>re is some x ∈ S X such that|x ∗ 0(x)| 1 − ε <str<strong>on</strong>g>and</str<strong>on</strong>g> ‖x + x 0 ‖ 2 − ε.(ii ∗ ) For all x 0 ∈ S X , x ∗ 0 ∈ S X ∗ <str<strong>on</strong>g>and</str<strong>on</strong>g> ε > 0, <strong>the</strong>re is some x ∗ ∈ S X ∗ such that|x ∗ (x 0 )| 1 − ε <str<strong>on</strong>g>and</str<strong>on</strong>g> ‖x ∗ + x ∗ 0‖ 2 − ε.((iii) B X = co T [ B X \ ( )] )x + (2 − ε)B X for every x ∈ S X <str<strong>on</strong>g>and</str<strong>on</strong>g> every ε > 0 (see Figure 4 below).(iii ∗ ) B X ∗ = co w∗( T [ B X ∗ \ ( x ∗ + (2 − ε)B X ∗)] ) for every x ∗ ∈ S X ∗ <str<strong>on</strong>g>and</str<strong>on</strong>g> every ε > 0.(iv) B X ∗ ⊕ ∞ X ∗∗ = cow∗ ( {(x ∗ , x ∗∗ ) : x ∗ ∈ ext(B X ∗), x ∗∗ ∈ ext(B X ∗∗), |x ∗∗ (x ∗ )| = 1} ) .It is clear that both spaces with <strong>the</strong> Daugavet property <str<strong>on</strong>g>and</str<strong>on</strong>g> spaces with <strong>numerical</strong> <strong>index</strong> 1 have <strong>the</strong>alternative Daugavet property. Both c<strong>on</strong>verses are false: <strong>the</strong> space c 0 ⊕ 1 C([0, 1], l 2 ) has <strong>the</strong> alternativeDaugavet property but fails <strong>the</strong> Daugavet property <str<strong>on</strong>g>and</str<strong>on</strong>g> its <strong>numerical</strong> <strong>index</strong> is not 1 [61, Example 3.2].Never<strong>the</strong>less, under certain isomorphic c<strong>on</strong>diti<strong>on</strong>s, <strong>the</strong> alternative Daugavet property forces <strong>the</strong> <strong>numerical</strong><strong>index</strong> to be 1.Propositi<strong>on</strong> 12 ([53, Remark 6]) Let X be a Banach space with <strong>the</strong> alternative Daugavet property. If Xhas <strong>the</strong> Rad<strong>on</strong>-Nikodým property or X is an Asplund space, <strong>the</strong>n n(X) = 1.With this result in mind, <strong>on</strong>e realizes that <strong>the</strong> necessary c<strong>on</strong>diti<strong>on</strong>s for a real Banach space to berenormed with <strong>numerical</strong> <strong>index</strong> 1 given in secti<strong>on</strong> 4 (namely Theorem 5 <str<strong>on</strong>g>and</str<strong>on</strong>g> Corollary 4), can be writtenin terms <strong>of</strong> <strong>the</strong> alternative Daugavet property. Even more, in <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Propositi<strong>on</strong> 7 given in [53],<strong>on</strong>ly rank-<strong>on</strong>e operators are used <str<strong>on</strong>g>and</str<strong>on</strong>g>, <strong>the</strong>refore, it can be also written in terms <strong>of</strong> <strong>the</strong> alternative Daugavetproperty.173
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. PayáxB X \ (−x + (2 − ε)B X )B X \ (x + (2 − ε)B X )Figure 4. l 2 ∞ has <strong>the</strong> alternative Daugavet propertyPropositi<strong>on</strong> 13 ([53, Lemma 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> Remark 6]) Let X be a Banach space with <strong>the</strong> alternative Daugavetproperty. Then,(a) |x ∗∗ (x ∗ )| = 1 for every x ∗∗ ∈ ext(B X ∗∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every weak ∗ -denting point x ∗ ∈ B X ∗.(b) |x ∗ (x)| = 1 for every x ∗ ∈ ext(B X ∗) <str<strong>on</strong>g>and</str<strong>on</strong>g> every denting point x ∈ B X .Propositi<strong>on</strong> 14 ([61, Remark 2.8]) Let X be an infinite-dimensi<strong>on</strong>al real Banach space with <strong>the</strong> alternativeDaugavet property. If X has <strong>the</strong> Rad<strong>on</strong>-Nikodým property, <strong>the</strong>n X c<strong>on</strong>tains l 1 . If X is an Asplundspace, <strong>the</strong>n X ∗ c<strong>on</strong>tains l 1 . In particular, X ∗∗ /X is not separable.The above two results give us an indicati<strong>on</strong> <strong>of</strong> why it is difficult to find characterizati<strong>on</strong>s <strong>of</strong> Banachspaces with <strong>numerical</strong> <strong>index</strong> 1 that do no involve operators. Indeed, it is not easy to c<strong>on</strong>struct n<strong>on</strong>compactoperators <strong>on</strong> an abstract Banach space. Thus, when <strong>on</strong>e uses <strong>the</strong> assumpti<strong>on</strong> that a Banach space has<strong>numerical</strong> <strong>index</strong> 1, <strong>on</strong>ly <strong>the</strong> alternative Daugavet property can be easily exploited. Of course, things areeasier if <strong>on</strong>e is working in a c<strong>on</strong>text where <strong>the</strong> alternative Daugavet property ensures <strong>numerical</strong> <strong>index</strong> 1,as it happens with Asplund spaces <str<strong>on</strong>g>and</str<strong>on</strong>g> spaces with <strong>the</strong> Rad<strong>on</strong>-Nikodým property. Therefore, it wouldbe desirable to find more isomorphic properties ensuring that <strong>the</strong> alternative Daugavet property implies<strong>numerical</strong> <strong>index</strong> 1. One possibility is <strong>the</strong> following.Problem 32 Let X be a Banach space which does not c<strong>on</strong>tain any copy <strong>of</strong> l 1 <str<strong>on</strong>g>and</str<strong>on</strong>g> having <strong>the</strong> alternativeDaugavet property. Is it true that n(X) = 1?An affirmative answer to <strong>the</strong> above questi<strong>on</strong> might come from <strong>the</strong> following more ambitious <strong>on</strong>e.Problem 33 Let X be a Banach space with <strong>the</strong> alternative Daugavet property. Is it true that v(T ) = ‖T ‖for every operator T ∈ L(X) fixing no copy <strong>of</strong> l 1 ?On <strong>the</strong> o<strong>the</strong>r h<str<strong>on</strong>g>and</str<strong>on</strong>g>, it is not possible to find isomorphic properties ensuring that <strong>the</strong> alternative Daugavetproperty <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> Daugavet property are equivalent.174
Numerical <strong>index</strong> <strong>of</strong> Banach spacesPropositi<strong>on</strong> 15 ([61, Corollary 3.3]) Let X be a Banach space with <strong>the</strong> alternative Daugavet property.Then <strong>the</strong>re exists a Banach space Y , isomorphic to X, which has <strong>the</strong> alternative Daugavet property butfails <strong>the</strong> Daugavet property.We may <strong>the</strong>n look for isometric c<strong>on</strong>diti<strong>on</strong>s that allow passing from <strong>the</strong> alternative Daugavet property to<strong>the</strong> Daugavet property. Having a complex structure could be such a c<strong>on</strong>diti<strong>on</strong>.Problem 34 Let X be a complex Banach space such that X R has <strong>the</strong> alternative Daugavet property. Doesit follow that X (equivalently X R ) has <strong>the</strong> Daugavet property?8. The polynomial <strong>numerical</strong> indicesIn 1968, <strong>the</strong> c<strong>on</strong>cept <strong>of</strong> <strong>numerical</strong> range <strong>of</strong> operators was extended to arbitrary c<strong>on</strong>tinuous functi<strong>on</strong>s from<strong>the</strong> unit sphere <strong>of</strong> a real or complex Banach space into <strong>the</strong> space by F. B<strong>on</strong>sall, B. Cain, <str<strong>on</strong>g>and</str<strong>on</strong>g> H. Schneider[8] in <strong>the</strong> obvious way. They showed that <strong>the</strong> <strong>numerical</strong> range <strong>of</strong> a bounded linear operator is alwaysc<strong>on</strong>nected, <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> same is true for arbitrary c<strong>on</strong>tinuous functi<strong>on</strong>s with <strong>on</strong>ly <strong>on</strong>e trivial excepti<strong>on</strong>: when <strong>the</strong>space has dimensi<strong>on</strong> <strong>on</strong>e over R. Three years later, L. Harris [33] studied various possible <strong>numerical</strong> rangesfor holomorphic functi<strong>on</strong>s <strong>on</strong> complex Banach spaces <str<strong>on</strong>g>and</str<strong>on</strong>g> got some deep results <strong>on</strong> <strong>the</strong>m.In this secti<strong>on</strong>, we c<strong>on</strong>centrate <strong>on</strong> homogeneous polynomials <strong>on</strong> real or complex spaces. Let us give <strong>the</strong>necessary definiti<strong>on</strong>s; we refer <strong>the</strong> reader to <strong>the</strong> book [19] by S. Dineen for background <strong>on</strong> polynomials.Given a Banach space X <str<strong>on</strong>g>and</str<strong>on</strong>g> a positive integer k, a mapping P : X −→ X is called a (c<strong>on</strong>tinuous)k-homogeneous polynomial <strong>on</strong> X if <strong>the</strong>re is an k-linear c<strong>on</strong>tinuous mapping A : X × · · · × X −→ Xsuch that P (x) = A(x, . . . , x) for every x ∈ X. A polynomial <strong>on</strong> X is a finite sum <strong>of</strong> homogeneouspolynomials. The space P(X; X) <strong>of</strong> all polynomials <strong>on</strong> X is normed by‖P ‖ = sup ‖P (x)‖x∈B X(P ∈ P(X; X)).We write P( k X; X) to denote <strong>the</strong> subspace <strong>of</strong> P(X; X) <strong>of</strong> those k-homogeneous polynomials <strong>on</strong> X, whichis a Banach space. The <strong>numerical</strong> range <strong>of</strong> P ∈ P(X; X) is <strong>the</strong> set <strong>of</strong> scalars<str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> <strong>numerical</strong> radius <strong>of</strong> P isV (P ) := {x ∗ (P (x)) : x ∈ S X , x ∗ ∈ S X ∗, x ∗ (x) = 1},v(P ) := sup{|λ| : λ ∈ V (P )}.As in <strong>the</strong> linear case, it is natural to define <strong>the</strong> polynomial <strong>numerical</strong> <strong>index</strong> <strong>of</strong> order k <strong>of</strong> X to be <strong>the</strong> c<strong>on</strong>stantn (k) (X) := inf { v(P ) : P ∈ P( k X; X), ‖P ‖ = 1 }= max { M 0 : ‖P ‖ Mv(P ) ∀P ∈ P( k X; X) } ;<strong>of</strong> course, n (1) (X) coincides with <strong>the</strong> usual <strong>numerical</strong> <strong>index</strong> <strong>of</strong> <strong>the</strong> space X. This definiti<strong>on</strong> was introducedvery recently by Y. Choi, D. García, S. Kim, <str<strong>on</strong>g>and</str<strong>on</strong>g> M. Maestre [13]. Note that 0 n (k) (X) 1, <str<strong>on</strong>g>and</str<strong>on</strong>g> thatn (k) (X) > 0 if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if v( · ) is a norm <strong>on</strong> P( k X; X) equivalent to <strong>the</strong> usual norm.The first result that we would like to menti<strong>on</strong> here is an extensi<strong>on</strong> <strong>of</strong> Glickfeld’s result for linear operators[26] given by L. Harris [33, Theorem 1]: for complex spaces, <strong>the</strong> <strong>numerical</strong> radius is always anequivalent norm in <strong>the</strong> space <strong>of</strong> k-homogeneous polynomials. More c<strong>on</strong>cretely, if X is a complex Banachspace <str<strong>on</strong>g>and</str<strong>on</strong>g> k 2, <strong>the</strong>n( ) k log(k)n (k) (X) exp. (9)1 − kIt was also proved in [33, §7] that <strong>the</strong> above inequalities are sharp. In <strong>the</strong> real case, Harris’ result above isfalse since <strong>the</strong> polynomial <strong>numerical</strong> indices <strong>of</strong> a real space may vanish (see Example 8.b below).The next result shows <strong>the</strong> behavior <strong>of</strong> <strong>the</strong> polynomial <strong>numerical</strong> <strong>index</strong> when <strong>the</strong> degree grows.175
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. PayáPropositi<strong>on</strong> 16 ([13, Propositi<strong>on</strong> 2.5]) Let X be a Banach space <str<strong>on</strong>g>and</str<strong>on</strong>g> let k be a positive integer. Thenn (k+1) (X) n (k) (X).Next we list several examples <strong>of</strong> Banach spaces for which <strong>the</strong>re is some informati<strong>on</strong> <strong>on</strong> <strong>the</strong>ir polynomial<strong>numerical</strong> indices.Examples 8 ([13], [14] <str<strong>on</strong>g>and</str<strong>on</strong>g> [15])(a) n (k) (K) = 1 for every k ∈ N.(b) If H is a real Hilbert space <strong>of</strong> dimensi<strong>on</strong> greater than 1, <strong>the</strong>n n (k) (H) = 0 for every k ∈ N.(c) If H is a complex Hilbert space <strong>of</strong> dimensi<strong>on</strong> greater than 1, <strong>the</strong>n 1/4 n (2) (H) 1/2 for everyk ∈ N.(d) In <strong>the</strong> complex case, all <strong>the</strong> polynomial <strong>numerical</strong> indices <strong>of</strong> an L 1 -predual are equal to 1. In particular,this is <strong>the</strong> case for <strong>the</strong> complex spaces c 0 , l ∞ <str<strong>on</strong>g>and</str<strong>on</strong>g>, more generally, C(K).(e) n (2) (l 1 ) 1/2 in <strong>the</strong> real as well as in <strong>the</strong> complex case.(f) For every k 2, <strong>the</strong> real spaces c 0 , l ∞ , c, <str<strong>on</strong>g>and</str<strong>on</strong>g> l m ∞ with m 2, have <strong>numerical</strong> <strong>index</strong> <strong>of</strong> order ksmaller than 1.(g) Actually, if K is a n<strong>on</strong>-perfect compact Hausdorff space with at least two points, <strong>the</strong>n <strong>the</strong> real spaceC(K) has <strong>numerical</strong> <strong>index</strong> <strong>of</strong> order k smaller than 1 for every k 2.The above examples show that <strong>the</strong> polynomial <strong>numerical</strong> indices distinguish between L-spaces <str<strong>on</strong>g>and</str<strong>on</strong>g> M-spaces in <strong>the</strong> complex case, <str<strong>on</strong>g>and</str<strong>on</strong>g> this is not possible if we <strong>on</strong>ly use <strong>the</strong> usual (linear) <strong>numerical</strong> <strong>index</strong>. Theyalso show that, unlike <strong>the</strong> linear case, <strong>the</strong>re is no relati<strong>on</strong>ship between <strong>the</strong> polynomial <strong>numerical</strong> indices <strong>of</strong>a Banach space <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> <strong>on</strong>es <strong>of</strong> its dual (see items (d) <str<strong>on</strong>g>and</str<strong>on</strong>g> (e) above). Never<strong>the</strong>less, <strong>the</strong>re is a relati<strong>on</strong>shipwith <strong>the</strong> polynomial <strong>numerical</strong> indices <strong>of</strong> <strong>the</strong> bidual.Propositi<strong>on</strong> 17 ([13, Corollary 2.15]) Let X be a Banach space. Then, we havefor every k 2.n (k) (X ∗∗ ) n (k) (X)We do not know if <strong>the</strong> above inequalities are actually equalities.Problem 35 Is <strong>the</strong>re a Banach space X such that n (2) (X ∗∗ ) < n (2) (X)?Let us present now more <str<strong>on</strong>g>open</str<strong>on</strong>g> <str<strong>on</strong>g>questi<strong>on</strong>s</str<strong>on</strong>g> arising from Examples 8.Problem 36 Compute <strong>the</strong> polynomial <strong>numerical</strong> indices <strong>of</strong> complex Hilbert spaces <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>of</strong> l 1 .Problem 37 Is n (2)( C[0, 1] ) = 1 in <strong>the</strong> real case? Is n (2)( L 1 [0, 1] ) = 1 in <strong>the</strong> real <str<strong>on</strong>g>and</str<strong>on</strong>g>/or in <strong>the</strong> complexcase?The behavior <strong>of</strong> polynomial <strong>numerical</strong> indices under direct sums <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> indices <strong>of</strong> vector-valued c<strong>on</strong>tinuousfuncti<strong>on</strong>s spaces were also studied in [13]. The results can be summarizedas follows.Propositi<strong>on</strong> 18 ([13, Propositi<strong>on</strong>s 2.8 <str<strong>on</strong>g>and</str<strong>on</strong>g> 2.10]) Let k be a positive integer.176
Numerical <strong>index</strong> <strong>of</strong> Banach spaces(a) If {X λ : λ ∈ Λ} is any family <strong>of</strong> Banach spaces, <strong>the</strong>nn (k)( [⊕ λ∈Λ X λ ] c0) infλ n(k) (X λ )n (k)( [⊕ λ∈Λ X λ ] l1) infλ n(k) (X λ )n (k)( [⊕ λ∈Λ X λ ] l∞) infλ n(k) (X λ ).(b) If X is a Banach space <str<strong>on</strong>g>and</str<strong>on</strong>g> K is a compact Hausdorff space, <strong>the</strong>nn (k)( C(K, X) ) n (k) (X).Examples 8 show that <strong>the</strong> inequalities in item (a) above are not always equalities in <strong>the</strong> real case; justtake Λ = N <str<strong>on</strong>g>and</str<strong>on</strong>g> X n = R for every n ∈ N. In <strong>the</strong> complex case, <strong>the</strong> same is true for <strong>the</strong> inequality involvingl 1 -sums (Λ = N <str<strong>on</strong>g>and</str<strong>on</strong>g> X n = C for every n ∈ N), but we do not know <strong>the</strong> answer for <strong>the</strong> o<strong>the</strong>r sums.Problem 38 Let {X λ : λ ∈ Λ} be a family <strong>of</strong> complex Banach spaces. Is is true thatn (k)( )[⊕ λ∈Λ X λ ] c0= n (k)( )[⊕ λ∈Λ X λ ] l∞= infλ n(k) (X λ ) ?For item (b) <strong>of</strong> Propositi<strong>on</strong> 18 <strong>the</strong> situati<strong>on</strong> is similar. For instance, <strong>the</strong> real space C({1, 2}, R) does nothave <strong>the</strong> same polynomial <strong>numerical</strong> indices as R. We do not know if <strong>the</strong>re exists an example <strong>of</strong> <strong>the</strong> samekind in <strong>the</strong> complex case.Problem 39 Let X be a complex Banach space. Is n (k)( C(K, X) ) equal to n (k) (X)?For spaces <strong>of</strong> Bochner-measurable integrable or essentially bounded vector valued functi<strong>on</strong>s no resultsare yet available.Problem 40 Let X be a Banach space, µ a positive σ-finite measure <str<strong>on</strong>g>and</str<strong>on</strong>g> k 2. Is <strong>the</strong>re any relati<strong>on</strong>shipbetween n (k)( L 1 (µ, X) ) (resp. n (k)( L ∞ (µ, X) ) ) <str<strong>on</strong>g>and</str<strong>on</strong>g> n (k) (X)?Item (a) in Propositi<strong>on</strong> 18 can be used to produce <strong>the</strong> following nice example.Example 9 There exists a complex Banach space X such that( ) k log(k)n (k) (X) = exp1 − kfor every k 2, i.e. all <strong>the</strong> inequalities in (9) are simultaneously equalities.PROOF. For each positive integer k 2, let X k be <strong>the</strong> two-dimensi<strong>on</strong>al complex Banach space givenin [33, §7] which satisfies <strong>the</strong> required equality for this k. Then, <strong>the</strong> spacesatisfiesX = [⊕ k2 X k ] c0n (k) (X) n (k) (X k ) = expby Propositi<strong>on</strong> 18.a, <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> o<strong>the</strong>r inequality always holds.( ) k log(k)1 − kAfter presenting <strong>the</strong> known results <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>open</str<strong>on</strong>g> <str<strong>on</strong>g>questi<strong>on</strong>s</str<strong>on</strong>g> related to <strong>the</strong> computing <strong>of</strong> <strong>the</strong> polynomial<strong>numerical</strong> indices, we would like to discuss <strong>the</strong> isometric or structural c<strong>on</strong>sequences that <strong>the</strong>se indices mayhave <strong>on</strong> a Banach space. Actually, we do not know <strong>of</strong> any result in this line, so we simply state c<strong>on</strong>jectures<str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>open</str<strong>on</strong>g> <str<strong>on</strong>g>questi<strong>on</strong>s</str<strong>on</strong>g>.First, we do not know what are precisely <strong>the</strong> values that <strong>the</strong> polynomial <strong>numerical</strong> <strong>index</strong> may take.177
V. Kadets, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> R. PayáProblem 41 For a given k 2, describe <strong>the</strong> sets{}n (k) (X) : X real Banach space <str<strong>on</strong>g>and</str<strong>on</strong>g>{}n (k) (X) : X complex Banach space .We may also ask for <strong>the</strong> class <strong>of</strong> Banach spaces for which <strong>the</strong> <strong>the</strong> k-order <strong>numerical</strong> <strong>index</strong> is <strong>on</strong>e <strong>of</strong> <strong>the</strong>extreme values.Problem 42 Study <strong>the</strong> real Banach spaces X satisfying n (2) (X) = 0. For instance, do <strong>the</strong>y satisfy thatn(X) = 0? What is <strong>the</strong> answer for finite-dimensi<strong>on</strong>al spaces?Problem 43 Characterize <strong>the</strong> complex Banach spaces X satisfying n (k) (X) = 1 for all k 2. Inparticular, do <strong>the</strong>y always c<strong>on</strong>tain a copy <strong>of</strong> c 0 in <strong>the</strong> infinite-dimensi<strong>on</strong>al case?Problem 44 Is <strong>the</strong>re any real Banach space X different from R such that n (2) (X) = 1?It would be desirable to get more informati<strong>on</strong> <strong>on</strong> <strong>the</strong> n<strong>on</strong>-increasing sequence {n (k) (X)} k∈N <strong>of</strong> allpolynomial <strong>numerical</strong> indices <strong>of</strong> a Banach space X. For instance, in view <strong>of</strong> <strong>the</strong> Examples 8, <strong>the</strong> followingquesti<strong>on</strong> arises.Problem 45 Is <strong>the</strong>re any Banach space X for which limk→∞ n(k) (X) ≠ 0, 1?To finish <strong>the</strong> secti<strong>on</strong>, let us menti<strong>on</strong> that Y. Choi, D. García, M. Martín, <str<strong>on</strong>g>and</str<strong>on</strong>g> M. Maestre have extended<strong>the</strong> study <strong>of</strong> <strong>the</strong> Daugavet equati<strong>on</strong> to polynomials <strong>on</strong> a Banach space in <strong>the</strong> very recent paper [14]. Givena Banach space X, a polynomial P ∈ P(X; X) satisfies <strong>the</strong> Daugavet equati<strong>on</strong> if<str<strong>on</strong>g>and</str<strong>on</strong>g> it satisfies <strong>the</strong> alternative Daugavet equati<strong>on</strong> if‖Id + P ‖ = 1 + ‖P ‖,max ‖Id + ω P ‖ = 1 + ‖P ‖.ω∈TDaugavet <str<strong>on</strong>g>and</str<strong>on</strong>g> alternative Daugavet properties are also translated in [14] to <strong>the</strong> polynomial setting by directgeneralizati<strong>on</strong> <strong>of</strong> <strong>the</strong> linear case. A Banach space X has <strong>the</strong> k-order Daugavet property (resp. <strong>the</strong> k-orderalternative Daugavet property) if every rank-<strong>on</strong>e k-homogeneous polynomial <strong>on</strong> X satisfies <strong>the</strong> Daugavetequati<strong>on</strong> (resp. <strong>the</strong> alternative Daugavet equati<strong>on</strong>). These properties are related to <strong>the</strong> polynomial <strong>numerical</strong>indices since, as in <strong>the</strong> linear case, for a polynomial P we have that(a) P satisfies <strong>the</strong> Daugavet equati<strong>on</strong> if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if ‖P ‖ = sup Re V (P ),(b) P satisfies <strong>the</strong> alternative Daugavet equati<strong>on</strong> if <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>on</strong>ly if ‖P ‖ = v(P )(see [14, Propositi<strong>on</strong> 1.3]). Let us remark that part <strong>of</strong> <strong>the</strong> informati<strong>on</strong> given in Examples 8 comes fromresults <strong>on</strong> <strong>the</strong> Daugavet <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> alternative Daugavet properties [14].We c<strong>on</strong>clude this paper by menti<strong>on</strong>ing <strong>the</strong> following ra<strong>the</strong>r surprising result.Propositi<strong>on</strong> 19 ([14, Propositi<strong>on</strong> 3.3 <str<strong>on</strong>g>and</str<strong>on</strong>g> Remark 3.6]) Let X be a Banach space <str<strong>on</strong>g>and</str<strong>on</strong>g> let k 2. Then,<strong>the</strong> Daugavet equati<strong>on</strong> <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> alternative Daugavet equati<strong>on</strong>s are equivalent in P( k X; X) in <strong>the</strong> followingtwo cases:(a) if <strong>the</strong> base field is C, or(b) if <strong>the</strong> base field is R <str<strong>on</strong>g>and</str<strong>on</strong>g> k is even.If k is odd, <strong>the</strong>n <strong>the</strong> Daugavet <str<strong>on</strong>g>and</str<strong>on</strong>g> <strong>the</strong> alternative Daugavet equati<strong>on</strong> are not equivalent in P( k R; R).Acknowledgement. The work <strong>of</strong> <strong>the</strong> first named author was partially supported by a fellowshipfrom <strong>the</strong> Alex<str<strong>on</strong>g>and</str<strong>on</strong>g>er-v<strong>on</strong>-Humboldt Stiftung. Sec<strong>on</strong>d <str<strong>on</strong>g>and</str<strong>on</strong>g> third authors are partially supported by SpanishMCYT project no. BFM2003-01681 <str<strong>on</strong>g>and</str<strong>on</strong>g> Junta de Andalucía grant FQM-185.178
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