Linear Algebra Done Right · Problem 7B.20

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · spectral-theorem

Problem

Suppose

T \in \mathcal{L}(V)

is normal and

U

is a subspace of

V

that is invariant under

T

. (a) Prove that

U^\perp

is invariant under

T

. (b) Prove that

U

is invariant under

T^*

. (c) Prove that

(T|_U)^* = (T^*)|_U

. (d) Prove that

T|_U \in \mathcal{L}(U)

and

T|_{U^\perp} \in \mathcal{L}(U^\perp)

are normal operators. This exercise can be used to give yet another proof of the complex spectral theorem (use induction on

\dim V

and the result that

T

has an eigenvector).