Linear Algebra Done Right · Problem 7A.20

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · normal-operators

Problem

Suppose

P \in \mathcal{L}(V)

is such that

P^2 = P

. Prove that the following are equivalent. (a)

P

is self-adjoint. (b)

P

is normal. (c) There is a subspace

U

V

such that

P = P_U