Assume we have a look to the space $L^2(S^1)$. An orthonormal basis of $L^2(S^1)$ is given by $p_n(x)=e^{2\pi in x}$ $(n\in\Bbb{Z})$. One can also have a look at the operator $S(p_n)=sgn(n)\cdot p_n$ (where $sng(n)$ is the sign of the integer (convention $sgn(0)=1$). This is a bounded operator. I want to show that the commutator operator of S and the multiplication opertor $M_n(g)=p_n\cdot g$ $(g\in L^2(S^1))$ is compact, in particular: $[S,M_n]$ is compact for each $n\in\mathbb{Z}$. I think one can use an argument that the image is finite dimensional or a norm limit of finite rank operators. Can someone help me?
Thanks a lot