2025/07/18 by Tomasz Szczepanski, Szczepanski, Tomasz
Mathematics · #Advanced Banach Space Theory #Approximation property #Banach space #Compact operator #Compact operator on Hilbert space #Finite-rank operator #Hilbert space #Holomorphic and Operator Theory #Invariant subspace problem #Nonlinear Differential Equations Analysis #Separable space #Strictly singular operator #Unbounded operator
paper · pdf · doi:10.48550/arxiv.2507.14297
openalex publication_date 2025/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
An operator T on a Banach space is said to be of chain N if there exist non-scalar operators S1,…,SN-1 and a non-zero compact operator K such that T ↔ S1 ↔ S2 ↔ …↔ SN-1 ↔ K, where A↔ 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 ℓp (1≤ p<∞) 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.