MAT 578 HOMEWORK 8 Due: 10/18/07
Solve three of the following problems.
32. Let Hbe a Hilbert space and W∈B(H) an invertible operator. Define θ:B(H)→
B(H) by θ(A) = W AW −1.
(a) Prove that θis an automorphism of the Banach algebra structure of B(H).
(b) Prove that θis self-adjoint (i.e. θ(A∗) = θ(A)∗for all A∈B(H)) if and only if
Wis a scalar multiple of a unitary operator.
33. Let (X, M) be a measurable space, and let f:X→Cbe a bounded Borel-measurable
function. Let µ1and µ2be σ-finite measures on M. For i= 1, 2, let Tibe the
multiplication operator on L2(X, M, µi) defined by f. Suppose that µ1and µ2are
mutually absolutely continuous. Prove that T1and T2are unitarily equivalent (i.e.
there exists a unitary operator W∈BL2(µ1), L2(µ2)such that W T1W∗=T2).
(Hint: use the Radon-Nikodym theorem.)
34. Let Abe a unital Banach algebra, and let a∈A. Define f:C→Aby f(z) = eza.
Prove that fis differentiable (in norm), and that f0(z) = af(z).
35. In this problem you will prove the Fuglede theorem: if Sand Tcommute, and Tis
normal, then Sand T∗commute. (In fact, the proof works in any unital C∗-algebra.
You won’t actually use the underlying Hilbert space in the proof.)
Let S,T∈B(H) with Tnormal. Suppose that ST =T S .
(a) Define f:C→B(H) by f(z) = eizT ∗S e−izT ∗. Prove that fis entire, i.e. that f
is differentiable in C.
(b) Let fbe as in part (a). Prove that f(z) = ei(zT ∗+zT )Se−i(zT ∗+zT ). Use this to
prove that kf(z)k=kSkfor all z∈C. Use Liouville’s theorem to conclude that
fis constant.
(c) Use the derivative of ffrom part (a) to prove that ST ∗=T∗S.
36. Let C(C) denote the (unital commutative ∗-) algebra of all continuous complex-valued
functions on C. For X⊆Ccompact, let k·kXbe the seminorm on C(C) defined by
kfkX= sup{|f(z)|:z∈X}.
(a) Let T∈B(H) be normal. Prove that there is a unique ∗-homomorphism f∈
C(C)7→ f(T)∈B(H) such that ζ7→ T(where ζ(z) = z), and such that
kf(T)k=kfkσ(T)for all f∈C(C).
(b) Let f∈C(C), and let Tn∈B(H) be a norm-convergent sequence of normal
operators, with limnTn=T. Prove that limnf(Tn) = f(T).
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