Linear Algebra Done Right · Problem 7F.30

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · polar-decomposition

Problem

Suppose

T \in \mathcal{L}(V)

S \in \mathcal{L}(V)

is a unitary operator, and

R \in \mathcal{L}(V)

is a positive operator such that

T = SR

. Prove that

R = \sqrt{T^*T}

. This exercise shows that if we write

T

as the product of a unitary operator and a positive operator (as in the polar decomposition 7.93), then the positive operator equals

\sqrt{T^*T}