On the chain of commuting operators on Banach spaces

Abstract An operator on a Banach space is said to be of chain if there exist non‐scalar operators and a non‐zero compact operator such that where denotes . We investigate this concept by identifying classes of operators that are of chain for some . Our main result establishes that every weighted shift on () 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.