Hi,(adsbygoogle = window.adsbygoogle || []).push({});

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

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!

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Proving Cayley Transform operator is unitary

Loading...

Similar Threads for Proving Cayley Transform |
---|

I Repetitive Fourier transform |

I Proving the Linear Transformation definition |

I Units of Fourier Transform |

I A problematic limit to prove |

A Help with Discrete Sine Transform |

**Physics Forums - The Fusion of Science and Community**