Isomorphisms and strictly singular operators in mixed Tsirelson spaces

TL;DR

This paper investigates the structure of isomorphisms and strictly singular operators in mixed Tsirelson spaces, establishing minimality and the existence of non-compact operators under certain conditions.

Contribution

It introduces new results on the sequential minimality of modified mixed Tsirelson spaces and the existence of strictly singular non-compact operators in specific subspaces.

Findings

Sequential minimality of certain modified mixed Tsirelson spaces.

Existence of strictly singular non-compact operators in subspaces.

Results depend on regularity conditions and specific families used in space construction.

Abstract

We study the family of isomorphisms and strictly singular operators in mixed Tsirelson spaces and their modified versions setting. We show sequential minimality of modified mixed Tsirelson spaces $T_M[(\mc{S}_n,\theta_n)]$ satisfying some regularity conditions and present results on existence of strictly singular non-compact operators on subspaces of mixed Tsirelson spaces defined by the families $(\mc{A}_n)_n$ and $(\mc{S}_n)_n$.