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A remark on the distortion of twisted sums

Jesús Suárez

Abstract

Odell and Schlumprecht solved the distortion problem by proving that the classic sequence spaces $\ell_p$ for $1<p<\infty$ admit an inevitable biorthogonal system. In particular, these spaces are arbitrarily distortable. Later, Maurey extended the result to asymptotic $\ell_p$-spaces while Tomczak-Jaegermann did likewise for the Schatten classes. We observe that the Kalton-Peck spaces $Z_p$ for $1<p<\infty$ admit an inevitable biorthogonal system. Therefore, we may add this classic family to the list of arbitrarily distortable spaces.

A remark on the distortion of twisted sums

Abstract

Odell and Schlumprecht solved the distortion problem by proving that the classic sequence spaces for admit an inevitable biorthogonal system. In particular, these spaces are arbitrarily distortable. Later, Maurey extended the result to asymptotic -spaces while Tomczak-Jaegermann did likewise for the Schatten classes. We observe that the Kalton-Peck spaces for admit an inevitable biorthogonal system. Therefore, we may add this classic family to the list of arbitrarily distortable spaces.
Paper Structure (3 sections, 1 theorem, 24 equations)

This paper contains 3 sections, 1 theorem, 24 equations.

Table of Contents

  1. Introduction
  2. Main
  3. Examples

Key Result

Theorem 1

Let $Z$ be a twisted sum of $\ell_p$ with $1<p<\infty$. If $Z$ satisfies the Principle of Small Perturbations, then $Z$ admits an inevitable biorthogonal system for every $0<\delta<1/2$. In particular, $Z$ is arbitrarily distortable.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Example 1
  • proof
  • Example 2
  • proof
  • Example 3
  • proof