Measure, Integration & Real Analysis, 2021a

Section 10B Spectrum 311
22 Suppose T is a self-adjoint operator on a complex Hilbert space. Prove that
(T + iI)(T − iI) −1
is a unitary operator.
[The function z ↦→ (z + i)(z − i) −1 maps R to {z ∈ C : |z| = 1}\{1}.
Thus this exercise provides another useful illustration of the analogies showing
(a) unitary ⇐⇒ {z ∈ C : |z| = 1};(b) self-adjoint ⇐⇒ R.]
For T a bounded operator on a Banach space, define e T by
e T =
∞
T
∑
k
k! .
k=0
23 (a) Prove that if T is a bounded operator on a Banach space V, then the infinite
sum above converges in B(V) and ‖e T ‖≤e ‖T‖ .
(b) Prove that if S, T are bounded operators on a Banach space V such that
ST = TS, then e S e T = e S+T .
(c) Prove that if T is a self-adjoint operator on a complex Hilbert space, then
e iT is unitary.
A bounded operator T on a Hilbert space is called a partial isometry if
‖Tf‖ = ‖ f ‖ for all f ∈ (null T) ⊥ .
24 Suppose (X, S, μ) is a σ-finite measure space and h ∈ L ∞ (μ). As usual, let
M h ∈B ( L 2 (μ) ) denote the multiplication operator defined by M h f = fh.
Prove that M h is a partial isometry if and only if there exists a set E ∈Ssuch
that h = χ E
.
25 Suppose T is an isometry on a Hilbert space. Prove that T ∗ is a partial isometry.
26 Suppose T is a bounded operator on a Hilbert space V. Prove that T is a partial
isometry if and only if T ∗ T = P U for some closed subspace U of V.
Measure, Integration & Real Analysis, by Sheldon Axler