Linear Algebra Done Right · Problem 7F.26

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · polar-decomposition

Problem

Suppose

T \in \mathcal{L}(V)

. Prove that

T

is invertible if and only if there exists a unique unitary operator

S \in \mathcal{L}(V)

such that

T = S\sqrt{T^*T}