Measure, Integration & Real Analysis, 2021a

336 Chapter 10 Linear Maps on Hilbert Spaces
EXERCISES 10D
1 Prove that if T is a compact operator on a nonzero Hilbert space, then ‖T‖ 2 is
an eigenvalue of T ∗ T.
2 Prove that if T is a self-adjoint operator on a nonzero Hilbert space V, then
‖T‖ = sup{|〈Tf, f 〉| : f ∈ V and ‖ f ‖ = 1}.
3 Suppose T is a bounded operator on a Hilbert space V and U is a closed subspace
of V. Prove that the following are equivalent:
(a) U is an invariant subspace for T.
(b) U ⊥ is an invariant subspace for T ∗ .
(c) TP U = P U TP U .
4 Suppose T is a bounded operator on a Hilbert space V and U is a closed subspace
of V. Prove that the following are equivalent:
(a) U and U ⊥ are invariant subspaces for T.
(b) U and U ⊥ are invariant subspaces for T ∗ .
(c) TP U = P U T.
5 Suppose T is a bounded operator on a nonseparable normed vector space V.
Prove that T has a closed invariant subspace other than {0} and V.
6 Suppose T is an operator on a Banach space V with dimension greater than 2.
Prove that T has an invariant subspace other than {0} and V.
[For this exercise, T is not assumed to be bounded and the invariant subspace is
not required to be closed.]
7 Suppose T is a self-adjoint compact operator on a Hilbert space that has only
finitely many distinct eigenvalues. Prove that T has finite-dimensional range.
8 (a) Prove that if T is a self-adjoint compact operator on a Hilbert space, then
there exists a self-adjoint compact operator S such that S 3 = T.
(b) Prove that if T is a normal compact operator on a complex Hilbert space,
then there exists a normal compact operator S such that S 2 = T.
9 Suppose T is a compact normal operator on a nonzero Hilbert space V. Prove
that there is a subspace of V with dimension 1 or 2 that is an invariant subspace
for T.
[If F = C, the desired result follows immediately from the Spectral Theorem for
compact normal operators. Thus you can assume that F = R.]
10 Suppose T is a self-adjoint compact operator on a Hilbert space and ‖T‖ ≤ 1 4 .
Prove that there exists a self-adjoint compact operator S such that S 2 + S = T.
Measure, Integration & Real Analysis, by Sheldon Axler