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Duals of Tirilman spaces have unique subsymmetric basic sequences

Stephen. J. Dilworth, Denka Kutzarova, Bünyamin Sarı, Svetozar Stankov

Abstract

The Tirilman spaces $Ti(p,γ)$, $1<p<\infty$, were introduced by Casazza and Shura as variations of the spaces constructed by Tzafriri. We prove that all subsymmetric basic sequences in the dual space $Ti^*(p,γ)$ are equivalent to its canonical subsymmetic but not symmetric basis.

Duals of Tirilman spaces have unique subsymmetric basic sequences

Abstract

The Tirilman spaces , , were introduced by Casazza and Shura as variations of the spaces constructed by Tzafriri. We prove that all subsymmetric basic sequences in the dual space are equivalent to its canonical subsymmetic but not symmetric basis.
Paper Structure (2 sections, 13 theorems, 27 equations)

This paper contains 2 sections, 13 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Spaces with a unique subsymmetric basic sequence

Key Result

Proposition 1

For every $1<p<\infty$ and $0<\gamma<1$, the canonical basis $(e_n)_{n=1}^\infty$ is $1$-dominated by every normalized block basis of $(e_n)_{n=1}^\infty$.

Theorems & Definitions (19)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • proof
  • Proposition 5: Sa
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 9 more