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.

  • $\begingroup$ @ThomasRot of course, since K is compact, T is bounded, then TK-KT is compact. $\endgroup$ – hjinghao Sep 14 '16 at 11:28
  • $\begingroup$ @ThomasRot Check?what? TK is compact, KT is compact, then of course TK-KT is compact... $\endgroup$ – hjinghao Sep 14 '16 at 11:54
  • $\begingroup$ @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 $K$ $\endgroup$ – hjinghao Sep 14 '16 at 14:13
  • $\begingroup$ @ThomasRot never mind, hope to discuss with u when u feel better $\endgroup$ – hjinghao Sep 14 '16 at 22:02

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