Kunze Linear Algebra · Problem 8.4.7

Inner Product Spaces

Mathematics — Methods · cayley-transform-unitary

Problem

Let

V

be a complex inner product space and

T

a self-adjoint linear operator on

V

. Show that (a)

\|\alpha + iT\alpha\| = \|\alpha - iT\alpha\|

for every

\alpha

V

. (b)

\alpha + iT\alpha = \beta + iT\beta

if and only if

\alpha = \beta

. (c)

I + iT

is non-singular. (d)

I - iT

is non-singular. (e) Now suppose

V

is finite-dimensional, and prove that

U = (I - iT)(I + iT)^{-1}

is a unitary operator;

U

is called the Cayley transform of

T

. In a certain sense,

U = f(T)

, where

f(x) = (1 - ix)/(1 + ix)