16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9.5.6

Operators on Inner Product Spaces

Mathematics — Linear Algebra · cayley-transform

Problem

Let

V

be a finite-dimensional complex inner product space and

U

a unitary operator on

V

such that

U\alpha = \alpha

implies

\alpha = 0

. Let

f(z) = i\frac{(1+z)}{(1-z)}, \qquad z \neq 1

and show that (a)

f(U) = i(I + U)(I - U)^{-1}

; (b)

f(U)

is self-adjoint; (c) for every self-adjoint operator

T

V

, the operator

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

is unitary and such that

T = f(U)