# Unitary Operators: Proving <Af,Ag>=<f,g>

• Physicsdudee
Thanks again.In summary, the person is trying to show that a unitary operator obeys <Af,Ag>=<f,g>, where A is a unitary operator. However, they are not given the fact that the adjoint of A is equal to its inverse, which is the problem. They have no clue how to prove the given task without using the mentioned statement.f

#### Physicsdudee

Homework Statement
Prove properties of unitary operator
Relevant Equations
Unitary maps are bijective and have the property of <Af,Ag>=<f,g>
Hello folks,

I need to show that a unitary operator obeys <Af,Ag>=<f,g>, where A is a unitary operator. However, I am technically not yet given the fact, that the adjoint of A is equal to its inverse, and that is the problem. I have no clue how to prove the given task without using the mentioned statement.
The space the scalaroduct is defined in is the $L^2$ space.

What is your definition of a unitary operator?

vanhees71
What is your definition of a unitary operator?
Well, my definiton of a unitary operator is the following: Let A be a unitary operator, then A is bijective and <Af,Ag>=<f,g> where f,g is an element of L2.

Well, my definiton of a unitary operator is the following: Let A be a unitary operator, then A is bijective and <Af,Ag>=<f,g> where f,g is an element of L2.
And what are you trying to prove?

vanhees71
And what are you trying to prove?
I need to prove one aspect of the given definition, namely the aspect that <Af,Ag>=<f,g>

I need to prove one aspect of the given definition, namely the aspect that <Af,Ag>=<f,g>
That's the definition you gave. You can't prove a definition!

vanhees71, vela and Orodruin
That's the definition you gave. You can't prove a definition!
Yeah, well I was thinking the same thing, that is why I was confused about it, the question itself seemed a bit unclear.

Yeah, well I was thinking the same thing, that is why I was confused about it, the question itself seemed a bit unclear.
There are generally two possible (and, of course, equivalent) definitions of a unitary operator.

1) It preserves the inner product.

2) Its adjoint is its inverse.

Whatever one you choose, you have to prove that the other is equivalent.

In fact, Wikipedia gives three equivalent definitions, but the thrid is more technical.

https://en.wikipedia.org/wiki/Unitary_operator

vanhees71
There are generally two possible (and, of course, equivalent) definitions of a unitary operator.

1) It preserves the inner product.

2) Its adjoint is its inverse.

Whatever one you choose, you have to prove that the other is equivalent.

In fact, Wikipedia gives three equivalent definitions, but the thrid is more technical.

https://en.wikipedia.org/wiki/Unitary_operator
Thanks for the help. I have read into the article already some time ago. I am noticing that I have misunderstood the assignment given, I think, at least the way I understand the assignment now makes more sense. I will give it a go.

vela