Measure, Integration & Real Analysis · Problem 10B.21

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · unitary-cayley-transform

Problem

Suppose

T

is a unitary operator on a complex Hilbert space such that

T - I

is invertible. Prove that

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

is a self-adjoint operator. [The function

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

maps

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

\mathbf{R}

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

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

and self-adjoint

\approx \mathbf{R}