Linear Algebra Done Right · Problem 7D.5

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Suppose

S \in \mathcal{L}(V)

. Prove that the following are equivalent. (a)

S

is a self-adjoint unitary operator. (b)

S = 2P - I

for some orthogonal projection

P

V

. (c) There exists a subspace

U

V

such that

Su = u

for every

u \in U

and

Sw = -w

for every

w \in U^\perp