Sequence-singular operators

TL;DR

This paper investigates collections of operators on Banach spaces, constructing uncountable chains of closed ideals in certain operator algebras, thereby resolving a longstanding open problem in the field.

Contribution

It introduces new types of operator collections that enable the construction of uncountable chains of closed ideals in specific Banach space operator algebras, answering a key open question.

Findings

Constructed uncountable chains of closed ideals in $\\mathcal{L}(\ell_1\oplus\ell_q)$ and $\\mathcal{L}(\ell_1\oplus c_0)$

Resolved a longstanding question of Pietsch regarding operator ideals

Developed new methods for analyzing operator collections on Banach spaces

Abstract

In this paper we study two types of collections of operators on a Banach space on the subject of forming operator ideals. One of the types allows us to construct an uncountable chain of closed ideals in each of the operator algebras $\mathcal{L}(\ell_1\oplus\ell_q)$, $1<q<\infty$, and $\mathcal{L}(\ell_1\oplus c_0)$. This finishes answering a longstanding question of Pietsch.