Measure, Integration & Real Analysis · Problem 10B.22

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · self-adjoint-cayley-transform

Problem

Suppose

T

is a self-adjoint operator on a complex Hilbert space. Prove that

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

is a unitary operator. [The function

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

maps

\mathbf{R}

\{z \in \mathbf{C} : |z| = 1\} \setminus \{1\}

. Thus this exercise provides another useful illustration of the analogies showing (a) unitary

\Longleftrightarrow \{z \in \mathbf{C} : |z| = 1\}

; (b) self-adjoint

\Longleftrightarrow \mathbf{R}