Linear Algebra Done Right · Problem 7D.8

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Prove or give a counterexample: If

S \in \mathcal{L}(V)

and there exists an orthonormal basis

e_1, \ldots, e_n

V

such that

\|Se_k\| = 1

for each

e_k

, then

S

is a unitary operator.