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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=(A-i)(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=(A-i)(A+i)^-1 so  (U)(x) = (A-i)((A+i)^-1)x  (U acting on an x)  Then (Ux,y)= {INTEGRAL}(A-i)((A+i)^-1)x y(conjugate) dx                           (1)  = {INTEGRAL}x(A-i)((A+i)^-1)(both conjugate)y(all three conjugate) dx         (2)  =(x,U*y)  and so deduce (U*)(y) = (A+i)((A-i)^-1)y  and so the adjoint of U is U*=(A+i)(A-i)^-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, $(AB)^*=B^*A^*$ and $(A^{-1})^*=(A^*)^{-1}$?

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 (A-i)(A+i)^-1 which is conjugated and then also the whole of (A-I)((A+i)^-1)y which is also conjugated.  Sorry if my notation is confusing, if unsure just ask.  Thanks for your help in advance!
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