SVD equivalent for infinite dimensional spaces

Question

How SVD can be extended to arbitrary linear operators? Any references?

Answer 1

It would make a great exercise to try to do it yourself in a simple case.

Here's an idea: let's suppose that $A$ is a compact operator on a Hilbert space. In particular, so are $A^* A$ and $A A^*$; these are also self adjiont, which by the spectral theorem implies that each has an orthonormal set of eigenvectors. Knowing what you do about the proof of the SVD in the finite dimensional case, can you think of a way to generalize that proof to this setting?

A good reference for the spectral theory of compact self adjoint operators is "Functional Analysis" by Lax.