On the chain of commuting operators on Banach spaces

Abstract

An operator $T$ on a Banach space is said to be of chain $N$ if there exist non-scalar operators $S_1,\dots ,S_{N-1}$ and a non-zero compact operator $K$ such that $$T\leftrightarrow S_1\leftrightarrow S_2\leftrightarrow \cdots \leftrightarrow S_{N-1}\leftrightarrow K,$$ where $A\leftrightarrow B$ denotes $AB=BA$. We investigate this concept by identifying classes of operators that are of chain $N$ for some $N$. Our main result establishes that every weighted shift on ${\ell }_p$ ($1\le p<\infty$) is of chain $3$, which in particular includes the class of non-Lomonosov operators studied by Hadwin et al. Furthermore, we provide an example of an operator on a separable Hilbert space that cannot be connected to a compact operator via a commuting chain of any length.

20 pages, accepted for publication in the Bulletin of the London Mathematical Society