Linear Algebra Done Right · Problem 7A.29

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · normal-operators

Problem

Prove or give a counterexample: If

T \in \mathcal{L}(V)

and there is an orthonormal basis

e_1, \ldots, e_n

V

such that

\|Te_k\| = \|T^* e_k\|

for each

k = 1, \ldots, n

, then

T

is normal.