MA4052 — Exercise Sheet 2: Final Homework Questions

12/12/12
MA4052 — Exercise Sheet 2: Final Homework Questions
1. Give an example of an operator T on a Hilbert space for which �T � �= r(T ), the
spectral radius of T
2. (a) Let H be a complex Hilbert space. Show that each T ∈ B(H) can be written
as T = T1 + iT2 for a unique choice of self-adjoint operators Ti ∈ B(H).
(b) Let T ∈ B(H). Show that T is normal if and only if T1 and T2 from (a)
commute.
(c) Let H = C 2 , with usual inner product, and define T ∈ B(H) by setting T x =
(x1 + ix2, x1 − ix2). Find T ∗ , and show that T ∗ T = T T ∗ = 2I. Find T1 and T2
from (a).
3. Let H be a Hilbert space, and R, S, T ∈ B(H).
(a) Show that for any α, β ∈ C with |α| = |β| we have that αR + βR ∗ is normal.
(b) Let S and T be normal operators satisfying ST ∗ = T ∗ S. Show that S ∗ T = T S ∗ ,
and that S + T and ST are normal.
4. Let H be a Hilbert space. A linear functional φ ∈ B(H) ∗ is called positive if φ(T ) � 0
whenever T ∈ B(H)+. A linear functional φ ∈ B(H) ∗ is called a state if it is positive
and φ(I) = 1.
(i) Show that S(H), the set of states, is a convex subset of B(H) ∗ , closed with
respect to the usual norm on B(H) ∗ .
(ii) Show that for each x ∈ H, the map ωx : B(H) → C given ωx(T ) = 〈x, T x〉
defines a positive linear functional. For which x is ωx a state?
5. Equip C n with its usual inner product structure: 〈z, w〉 = � n
i=1 ziwi, and denote the
resulting norm by �z� = � 〈z, z〉, leaving off the usual subscript. The operators in
T ∈ B(C n ) are naturally identified with matrices in Mn(C) by T ←→ [T i j ], where T i j =
〈ei, T ej〉 for the usual (orthonormal) basis {ei} n i=1 of C n . This question investigates
three norms on B(C n ), one of which is the standard operator norm:
�T �∞ := sup{�T x� : x ∈ C n , �x� � 1}.
Here we add the subscript ∞ to distinguish this norm from the other two introduced
below.
(a) Recall that the trace on Mn(C) is define by
tr(A) =
n�
A i i.
Show that tr is a linear functional on Mn(C) and that tr(AB) = tr(BA) for all
A, B ∈ Mn(C n ).
(b) Show that the following defines an inner product on B(C n ):
i=1
〈S, T 〉 = tr(S ∗ T ),

where S and T are identified with their matrices with respect to the usual orthonormal
basis of C n .
Denote the resulting norm on B(C n ) as follows:
(c) Define the following:
�T �2 := � 〈T, T 〉.
�T �1 := tr(|T |), T ∈ B(C n ),
where |T | denotes the positive part of T . By carrying out the following steps, show
that this defines a norm on B(C n ).
(i) If S � 0, then S is self-adjoint, hence is diagonalisable. Show that �S�1 is
the sum of the eigenvalues of S, and hence that �S�1 = 0 if and only if S = 0.
Hence show that for a general T ∈ B(C n ), �T �1 � 0, with equality if and
only if T = 0.
(ii) Show that �λT �1 = |λ|�T �1 for all T ∈ B(C n ) and λ ∈ C.
(iii) Show that for any R, T ∈ B(C n ) with T � 0 we have
0 � R ∗ T R � �T �∞R ∗ R.
(iv) Show that for all S, T ∈ B(C n ) we have
| tr(ST )| � �S�∞�T �1.
Hint: write ST = SU|T | 1/2 |T | 1/2 and use the Cauchy-Schwarz inequality.
(v) Show that for all S, T ∈ B(C n ) we have
�S + T �1 � �S�1 + �T �1.
Hint: write S + T = W |S + T | for some partial isometry W . Explain why
W ∗ (S + T ) = |S + T |. Why must we have �W ∗ �∞ � 1?
(d) For each T ∈ B(C n ) define a map ϕT : B(C n ) → C by
ϕT (S) = tr(ST ).
Show that ϕT is a linear functional on B(Cn ), and that if we consider ϕT as an
element of � B(Cn �∗ ), � · �∞ then
�ϕT � = �T �1.
Hence show that the map T ↦→ ϕT gives a linear, isometric isomorphism between
� � n
B(C , � · �1 and
� � n ∗.
B(C ), � · �∞
Finally, show that ψ ∈ � B(C n ), � · �∞
� ∗ is a state if and only if ψ = ϕT for some
T ∈ B(C n ) such that T � 0 and tr(T ) = 1.

When H is an infinite-dimensional Hilbert space, � �∗ B(H), � · �∞ is isometrically
isomorphic to the space of trace-class operators on H, which is an ideal in B(H)
consisting of those operators for which a sensible definition of trace can be given.
Note that in the finite-dimensional case, tr(I) = dim H, so for infinite-dimensional
H the identity operator will not be trace-class. The positive trace-class operators of
trace one (called density matrices) then correspond to states on B(H), which are of
great interest in quantum mechanics. The states in part (ii) of question 4 are known
as vector states, but there are many other mixed states. It can be shown, however,
that S(H), the set of all states, is the smallest closed convex set that contains all the
vector states, so the vector states in some sense generates S(H).
For the commutative counterpart, if X = C0(R; C), the space of C-values, bounded,
continuous functions on R that vanish at infinity ( lim
x→±∞ g(x) = 0), and if f ∈ L1 (R; C)
satisfies
�
f(x) � 0 for a.a. x ∈ R, f(x) dx = 1
then the following defines a state on X:
�
ϕf(g) =
R
R
g(x)f(x) dx,
in the sense that if g � 0 then ϕf(g) � 0, and ϕf(1) = 1, where 1 is considered both
as a number and as a constant function on R.
The classification of all bounded linear functionals on X is the subject of another
version of the Riesz Representation Theorem; the function f above is an example of
a probability density function, but not all states on X arise this way. For example, if
we pick any subset {x1, . . . , xn} ⊂ R and λ1, . . . , λn ∈ [0, ∞) with � n
i=1 λi = 1, then
ϕ(g) =
n�
λig(xi)
i=1
is also a state on X, but there is no f ∈ L1 (R) such that ϕ = ϕf.
Noncommutative versions of classical mathematical theories (e.g. quantum probability,
noncommutative geometry, quantum groups) take a classical theory, encode
it as a commutative algebra, then throw away the commutativity. For probability
theory, the basic set-up is a probability space (Ω, F, P), where Ω is the sample space,
F a σ-algebra of subsets, and P a probability measure. The algebra L∞ (Ω) consists
of random variables, and we have a state on L∞ (Ω) defined by
�
E[X] = X dP.
A particular example involves defining a probability measure on R using probability
density functions as above. But note that events, i.e. subsets A ∈ F, are associated
to the random variable 1A, where
1A(ω) =
Ω
�
1 if ω ∈ A,
0 if ω /∈ A.
Note also f ∈ L ∞ (Ω) satisfies f 2 = f if and only if f = 1A for some A ∈ F,
with P(A) = E[1A]. That is, events are projections in the algebra L ∞ (Ω), and the

probability of an event can be computed using the state. So probability theory is
encoded by an algebra A with a state ω. If we drop commutativity, then we can, for
example, consider (B(H), ωx) as a probability space, where elements of B(H) now
are the random variables, with the (orthogonal) projections corresponding to events!
But there are now plenty of questions to answer, for example given projections P and
Q, how do we define the event that P and Q both occur?