An Ordinal Index on the Space of Strictly Singular Operators

TL;DR

This paper introduces an ordinal index for strictly singular operators between separable Banach spaces, providing conditions under which this index is bounded, and applies it to various classes of Banach spaces.

Contribution

It defines a new ordinal index on strictly singular operators and establishes a bound under certain conditions, advancing the understanding of operator structure in Banach spaces.

Findings

Index is bounded by ω₁ under specific conditions

Application to totally incomparable and hereditarily indecomposable spaces

Provides a framework for analyzing operators with few operators

Abstract

Using the notion of $S_\xi$-strictly singular operator introduced by Androulakis, Dodos, Sirotkin and Troitsky, we define an ordinal index on the subspace of strictly singular operators between two separable Banach spaces. In our main result, we provide a sufficient condition implying that this index is bounded by $\omega_1$. In particular, we apply this result to study operators on totally incomparable spaces, hereditarily indecomposable spaces and spaces with few operators.