Linear Algebra Done Right · Problem 7F.24

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · operator-norms-unitary

Problem

Suppose

T \in \mathcal{L}(V)

is invertible. Prove that

\|T^{-1}\| = \|T\|^{-1} \iff \frac{T}{\|T\|} \text{ is a unitary operator.}