Let $\ell^2 = \ell^2(\mathbb{Z})$ denote the Hilbert space of square summable complex sequences on $\mathbb{Z}$ and suppose that $\sigma:\mathbb{Z} \to \mathbb{Z}$ is a function such that the linear operator $C_\sigma: \ell^2 \to \ell^2$ defined by $C_\sigma(\{\alpha_n\}) = \{\alpha_{\sigma(n)}\}$ is bounded. In other words, let $C_\sigma$ be the composition operator induced by $\sigma$.
Now let $F$ denote the isomorphism of $L^2(-\pi,\pi) \to \ell^2$ given by Fourier series. Then every operator $T:L^2(-\pi,\pi) \to L^2(-\pi,\pi)$ naturally induces an operator $\hat{T}: \ell^2 \to \ell^2$ given by $\hat{T} = FTF^{-1}$. My question is now the following:
Given $\sigma: \mathbb{Z} \to \mathbb{Z}$, is there a practical way of recovering the operator $T$ such that $\hat{T} = C_\sigma$? Is there a characterization of the operators $T$ such that $\hat{T} = C_\sigma$ for some $\sigma$?
I have tried simply plugging into the definition of Fourier series and working from there, but this has not yielded much help. Oh also, if someone can think of a better title, please edit.
Thanks in advance.