Linear Algebra Done Right · Problem 7D.16

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Suppose

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

is self-adjoint. Prove that

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

is a unitary operator and 1 is not an eigenvalue of this operator.