Measure, Integration & Real Analysis · Problem 10B.8

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · orthogonal-projections-self-adjoint

Problem

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

V

such that

P = P_U