Small operator ideals on the Schlumprecht and Schreier spaces

TL;DR

This paper develops a method to construct many distinct small operator ideals on specific Banach spaces, revealing a rich structure of these ideals on Schlumprecht and Schreier spaces.

Contribution

It introduces a new method for building operators that generate a vast number of distinct small closed operator ideals on these spaces.

Findings

Exactly 2^c small closed operator ideals on Schlumprecht space.

A chain of size c of small closed operator ideals on Schreier spaces of finite order.

Demonstrates the abundance of operator ideals in these Banach spaces.

Abstract

We present a method of building operators on a Banach space $X$ that generate distinct operator ideals in the algebra $\mathscr{B}(X)$ of bounded linear operators on $X$. We show that there are exactly $2^\mathfrak{c}$ distinct small closed operator ideals on the Schlumprecht space and there is a chain of cardinality $\mathfrak{c}$ of small closed operator ideals on any Schreier space of finite order.