On the chain of commuting operators on Banach spaces and Lomonosov's invariant subspace theorem

TL;DR

This paper explores the structure of commuting operator chains on Banach spaces, linking them to Lomonosov's invariant subspace theorem, and demonstrates that weighted shifts are of chain 3, with some operators not connected to compact operators.

Contribution

It introduces the concept of operator chains on Banach spaces, connects this to invariant subspace theory, and classifies weighted shifts as chain 3 operators, extending previous results.

Findings

Weighted shifts are of chain 3.

Operators from Hadwin et al. are of chain 3.

Existence of operators not connected to compact operators via any chain.

Abstract

An operator $T$ on a Banach space is said to be of chain $N$ if there exist non-scalar operators $S_1,...,S_{N-1}$ and a non-zero compact $K$ such that $$T \leftrightarrow S_1 \leftrightarrow S_2 \leftrightarrow ...\leftrightarrow S_{N-1} \leftrightarrow K,$$ where $A\leftrightarrow B$ means $AB=BA$. A connection of this theory to the Lomonosov's Invariant Subspace Theorem is highlighted. It is shown that for every weighted shift $T$ it is of chain $3$. In particular, every non-Lomonosov operator from from the work of Hadwin et al. is of chain $3$. An example of an operator on a separable Hilbert space is given, such that it fails to be connected to a compact operator via a chain of any length.