Linear Algebra Done Right · Problem 7F.3

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · operator-norms

Problem

Suppose

T \in \mathcal{L}(V, W)

and

v \in V

. Prove that

\|Tv\| = \|T\| \|v\| \iff T^*Tv = \|T\|^2 v.