Let $H=\ell^2$ be a Hilbert space with basis $\{\eta_i\}$. Let $X= \sum_{i=1}^\infty \frac{1}{i}\langle\cdot,\eta_i\rangle\eta_{i+1}$ and $T=\sum_{i=1}^\infty \langle\cdot,\eta_{i+1}\rangle\eta_{i}$. The question is whether there is a compact operator $K\in K(H)$ such that $$(T-K)X =X(T-K). $$
My idea is, $X$ is generated by right shift operator, so, $X$ only commutes with the convolution operators. But $T$ is left shift operator, and $K$ is compact, it looks like that $T-K$ could not be the convolution operator of right shift operator. But I don't know how to prove it clearly.