On the Characters and the Plancherel Formula of Nilpotent Groups ...

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 313
Proof. Lemma 6.0 implies that
v, = w, = cnb,b2 * ** b, for all n.
Let B be the unit ball E({l)) in H. Then
V,(B) = W,(B) = c, .
Using (5.11) and (5.12) the proof is complete.
PROPOSITION 6.2. For any compact ellipsoid E = E({b,}),
Thus if EV(E) ( - +, E is volumetric.
Proof.3 We have Y > A by Propositions 5.8 and 6.1, and r < h by
Proposition 5.7. The second conclusion follows then from 6.1 and
the definition of “volumetric” (just before Conjecture 5.9).
PROPOSITION 6.3. The following are equivalent :
(a) E = E({b,)) is a GC-set
(b) E is a GB-set
(c) C& bn2 < co (E is a “Schmidt ellipsoid”).
Proof. (a) implies (b) clearly if E is compact; if not, both fail.
If (b) holds, and A is the linear operator such that A(y,) = b,v,, ,
L o A has a version continuous on H (Theorem 4.3(e) above). It
is known that this is true if and only if A is a Hilbert-Schmidt oper-
ator (see [8], Lemma 4, p. 344). Thus (b) and (c) are equivalent.
Next, assume (c). Then for some k, 7 GO, C kn2bn2 < CO. Let
El = E {k,b,)). Th en E is El-compact and not maximal, so by
Theorem 4.7, E is a GC-set. Q.E.D.
It follows immediately from the above results that Conjectures
3.3, 5.4, and 5.9 all hold for ellipsoids.
Now we turn to our second class of examples. Let {Fm} be an
increasing sequence of subspaces of H with F, n-dimensional,
n = 0, 1, 2 ,... . Let b, 1 0. Specializing [14], we define the full
approximation set A = A({b,}) as
{X : for all 71, 11 x - yn I/ < b, for some y, inFn}.
s r = I\ is also proved by Prosser; see op. cit. in previous footnote.