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arXiv · 2507.14297

On the chain of commuting operators on Banach spaces

Tomasz Szczepanski
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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 $$T \leftrightarrow S_1 \leftrightarrow S_2 \leftrightarrow \dots\leftrightarrow S_{N-1} \leftrightarrow K,$$ 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 $\ell_p$ ($1\leq p<\infty$) 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.

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Tomasz Szczepanski. 2025-07-18. On the chain of commuting operators on Banach spaces. https://arxiv.org/abs/2507.14297

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