Measure, Integration & Real Analysis · Problem 10B.20

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · unitary-operators-characterization

Problem

Suppose

T

is a bounded operator on a complex Hilbert space, with

T = A + iB

, where

A

and

B

are self-adjoint (see 10.54). Prove that

T

is unitary if and only if

T

is normal and

A^2 + B^2 = I

. [If

z = x + yi

, where

x, y \in \mathbf{R}

, then

|z| = 1

if and only if

x^2 + y^2 = 1

. Thus this exercise strengthens the analogy between the unit circle in the complex plane and the unitary operators.]