Linear Algebra Done Right · Problem 7D.15

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Suppose

T

is a unitary operator on

V

such that

T - I

is invertible. (a) Prove that

(T + I)(T - I)^{-1}

is a skew operator (meaning that it equals the negative of its adjoint). (b) Prove that if

\mathbf{F} = \mathbf{C}

, then

i(T + I)(T - I)^{-1}

is a self-adjoint operator. The function

z \mapsto i(z+1)(z-1)^{-1}

maps the unit circle in

\mathbf{C}

(except for the point 1) to

\mathbf{R}

. Thus (b) illustrates the analogy between the unitary operators and the unit circle in

\mathbf{C}

, along with the analogy between the self-adjoint operators and

\mathbf{R}