Linear Algebra Done Right · Problem 7C.15

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · positive-operators

Problem

Suppose

T \in \mathcal{L}(V)

is self-adjoint. Prove that there exist positive operators

A, B \in \mathcal{L}(V)

such that

T = A - B \quad \text{and} \quad \sqrt{T^*T} = A + B \quad \text{and} \quad AB = BA = 0.