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

[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.

 

[PeroK]

What is your definition of a unitary operator?

 

[Physicsdudee]

What is your definition of a unitary operator?

Well, my definition 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.

 

[PeroK]

Well, my definition 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?

 

[Physicsdudee]

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>

 

[PeroK]

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!

 

[Physicsdudee]

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.

 

[PeroK]

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

 

[Physicsdudee]

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.