Hi Ad123q!
Quote by Ad123q
Hi,
Was wondering if anyone could give me a hand.
I need to prove that the Cayley Transform operator given by U=(Ai)(A+i)^1 is UNITARY, ie that UU*=U*U=I where U* is the adjoint of U (I am given also that A=A* in the set of bounded operators over a Hilbert space H).
My solution so far, is this correct?
U=(Ai)(A+i)^1 so
(U)(x) = (Ai)((A+i)^1)x (U acting on an x)
Then (Ux,y)= {INTEGRAL}(Ai)((A+i)^1)x y(conjugate) dx (1)
= {INTEGRAL}x(Ai)((A+i)^1)(both conjugate)y(all three conjugate) dx (2)
=(x,U*y)
and so deduce (U*)(y) = (A+i)((Ai)^1)y
and so the adjoint of U is U*=(A+i)(Ai)^1
It can then be checked that UU*=U*U=I

How do you conclude this from your expression for U*?
Btw, instead of using the integral, can't you simply use the properties of the adjoint operator?
That is, [itex](AB)^*=B^*A^*[/itex] and [itex](A^{1})^*=(A^*)^{1}[/itex]?
Quote by Ad123q
As you can see my main query is the mechanism of finding the adjoint of U for the given U.
For clarity in step (1) it is just the y which is conjugated, and in step (2) it is (Ai)(A+i)^1 which is conjugated and then also the whole of (AI)((A+i)^1)y which is also conjugated. Sorry if my notation is confusing, if unsure just ask.
Thanks for your help in advance!
