Strictly singular non-compact operators on a class of HI spaces

TL;DR

This paper introduces a method for constructing bounded strictly singular non-compact operators on certain complex Banach spaces, including mixed Tsirelson and hereditarily indecomposable spaces, expanding understanding of operator behavior in these spaces.

Contribution

It provides a novel construction technique for strictly singular non-compact operators on mixed Tsirelson and HI spaces, which was previously not well-understood.

Findings

Constructed explicit examples of such operators.

Extended the class of spaces known to admit these operators.

Enhanced understanding of operator structure in complex Banach spaces.

Abstract

We present a method for constructing bounded strictly singular non-compact operators on mixed Tsirelson spaces defined either by the families (A_n) or (S_n) of a certain class, as well as on spaces built on them, including hereditarily indecomposable spaces.