Proving Cayley Transform operator is unitary

In summary, the conversation discusses the proof of the UNITARY property of the Cayley Transform operator, given by U=(A-i)(A+i)^-1, and its adjoint operator U*. The process involves using the properties of the adjoint operator and the commutativity of the operators involved. The conversation also brings up alternative methods of proof using the properties of the adjoint operator.
  • #1
Ad123q
19
0
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

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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  • #2
Hi Ad123q! :smile:


Ad123q said:
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, [itex](AB)^*=B^*A^*[/itex] and [itex](A^{-1})^*=(A^*)^{-1}[/itex]?


Ad123q said:
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!
 
  • #3
[(A-i)(A+i)-1]* = [(A+i)-1]*(A-i)* = [(A+i)*]-1(A-i)*
=(A* - i)-1(A*+i) = (A - i)-1(A+i)

Now for instance multiply this with the original operator

(A - i)-1(A+i)(A-i)(A+i)-1

Note that A+i and A-i commute hence you get the result. Similiarly for the other way around
 

1. What is the Cayley Transform operator?

The Cayley Transform operator is a mathematical transformation that maps elements from one vector space to another. It is commonly used in linear algebra and is known for its ability to preserve the geometry and structure of the original space.

2. How is the Cayley Transform operator defined?

The Cayley Transform operator is defined as (I-A)(I+A)^-1, where I is the identity matrix and A is an invertible matrix. It can also be written as (I+A)(I-A)^-1.

3. What does it mean for the Cayley Transform operator to be unitary?

A unitary operator is one that preserves the inner product, or the angle and length of vectors, in a vector space. For the Cayley Transform operator, this means that it preserves the geometric structure of the original space.

4. How can you prove that the Cayley Transform operator is unitary?

To prove that the Cayley Transform operator is unitary, you can show that it satisfies the properties of a unitary operator. This includes showing that it is linear, that it preserves the inner product of vectors, and that its inverse is equal to its adjoint.

5. Why is it important to prove that the Cayley Transform operator is unitary?

Proving that the Cayley Transform operator is unitary is important because it confirms that it is a valid mathematical transformation. It also allows us to use the operator in various applications, such as solving differential equations, without worrying about distorting the underlying geometry of the vector space.

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