Linear Algebra Done Right · Problem 7B.19

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · spectral-theorem

Problem

Suppose

T \in \mathcal{L}(V)

is self-adjoint and

U

is a subspace of

V

that is invariant under

T

. (a) Prove that

U^\perp

is invariant under

T

. (b) Prove that

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

is self-adjoint. (c) Prove that

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

is self-adjoint.