How Do Continuous Linear Representations of S^1 Function in Hilbert Spaces?

[HMY]

Let H be a separable Hilbert space. What are the continuous
linear representations of S^1 on H?

I read in an article this is defined as in the finite-dim case.
Why is this so?

Thanks.

 

[morphism]

Well S^1 is compact, so every continuous linear representation of S^1 on H is unitary (re-norm H if necessary) and decomposes into a direct sum of irreducible representations. And since S^1 is abelian, its irreducible representations are nothing other than its characters, i.e. they're one-dimensional.

Is this what you're asking?