# commutator of a bounded linear operator on hilbert space

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.

• @ThomasRot of course, since K is compact, T is bounded, then TK-KT is compact. – hjinghao Sep 14 '16 at 11:28
• @ThomasRot Check?what？ TK is compact, KT is compact, then of course TK-KT is compact... – hjinghao Sep 14 '16 at 11:54
• @ThomasRot Well。。。 X is also compact, so the necessary condition is satisfied but what about the sufficiency...actually, I want to show to there is no such a $K$ – hjinghao Sep 14 '16 at 14:13
• @ThomasRot never mind, hope to discuss with u when u feel better – hjinghao Sep 14 '16 at 22:02