Linear Algebra Done Right · Problem 7A.12

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · normal-operators

Problem

An operator

B \in \mathcal{L}(V)

is called skew if

B^* = -B.

Suppose that

T \in \mathcal{L}(V)

. Prove that

T

is normal if and only if there exist commuting operators

A

and

B

such that

A

is self-adjoint,

B

is a skew operator, and

T = A + B