If T is a selfadjoint operator in a Hilbert space show that

If T is a self-adjoint operator in a Hilbert space, show that (T - iI)(T + iI)^-i (Cayley transform of T) is a unitary operator.

Solution

Given that T is a adjoint operator then

T*=T

Given ( T- iI) (T+ iI)^-1

let (T - iI) (T + iI)^-1 = A

   multiply bothsides with (T + iI) on right side then

   (T - iI) (T+ iI)^-1 (T + iI) = A(T + iI)

   (T - iI) I = AT +AiI

   T -iI =AT + AiI

subtract bothsides AT on left side

   AT - T -iI = AT -AT+AiI

   AT - T - iI = AiI

add bothsides iI on right side

   AT - T - iI+ iI = AiI+iI

AT - T =A+iI

(A - iI) T = A+ iI

   (A - iI) =(A+iI) T*

   (A - iI) =(A+iI) T

multiply both side with (A+iI)^-1  

   (A-iI) (A+iI)^-1 = T

Then we can say that T is unitory operator