Measure, Integration & Real Analysis, 2021a

Section 10B Spectrum 309
EXERCISES 10B
1 Verify all the assertions in Example 10.33.
2 Suppose T is a bounded operator on a Hilbert space V.
(a) Prove that sp(S −1 TS) =sp(T) for all bounded invertible operators S on V.
(b) Prove that sp(T ∗ )={α : α ∈ sp(T)}.
(c) Prove that if T is invertible, then sp(T −1 )= { 1
α
: α ∈ sp(T) } .
3 Suppose E is a bounded subset of F. Show that there exists a Hilbert space V
and T ∈B(V) such that the set of eigenvalues of T equals E.
4 Suppose E is a nonempty closed bounded subset of F. Show that there exists
T ∈B(l 2 ) such that sp(T) =E.
5 Give an example of a bounded operator T on a normed vector space such that
for every α ∈ F, the operator T − αI is not invertible.
6 Suppose T is a bounded operator on a complex nonzero Banach space V.
(a) Prove that the function
α ↦→ ϕ ( (T − αI) −1 f )
is analytic on C \ sp(T) for every f ∈ V and every ϕ ∈ V ′ .
(b) Prove that sp(T) ̸= ∅.
7 Prove that if T is an operator on a Hilbert space V such that 〈Tf, g〉 = 〈 f , Tg〉
for all f , g ∈ V, then T is a bounded operator.
8 Suppose P is a bounded operator on a Hilbert space V such that P 2 = P. Prove
that P is self-adjoint if and only if there exists a closed subspace U of V such
that P = P U .
9 Suppose V is a real Hilbert space and T ∈B(V). The complexification of T is
the function T C : V C → V C defined by
T C ( f + ig) =Tf + iTg
for f , g ∈ V (see Exercise 4 in Section 8B for the definition of V C ).
(a) Show that T C is a bounded operator on the complex Hilbert space V C and
‖T C ‖ = ‖T‖.
(b) Show that T C is invertible if and only if T is invertible.
(c) Show that (T C ) ∗ =(T ∗ ) C .
(d) Show that T is self-adjoint if and only if T C is self-adjoint.
(e) Use the previous parts of this exercise and 10.49 and 10.38 to show that if
T is self-adjoint and V ̸= {0}, then sp(T) ̸= ∅.
Measure, Integration & Real Analysis, by Sheldon Axler