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Norm attaining operators into locally asymptotically midpoint uniformly convex Banach spaces

Audrey Fovelle

Abstract

We prove that if $Y$ is a locally asymptotically midpoint uniformly convex Banach space which has either a normalized, symmetric basic sequence that is not equivalent to the unit vector basis in $\ell_1$, or a normalized sequence with upper p-estimates for some $p>1$, then $Y$ does not satisfy Lindenstrauss' property B.

Norm attaining operators into locally asymptotically midpoint uniformly convex Banach spaces

Abstract

We prove that if is a locally asymptotically midpoint uniformly convex Banach space which has either a normalized, symmetric basic sequence that is not equivalent to the unit vector basis in , or a normalized sequence with upper p-estimates for some , then does not satisfy Lindenstrauss' property B.
Paper Structure (7 sections, 9 theorems, 11 equations)

This paper contains 7 sections, 9 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Locally AMUC Banach spaces
  4. A family of Banach spaces
  5. Symmetric basic sequences and upper $p$-estimates
  6. Results
  7. Open questions

Key Result

Theorem A

Let $Y$ be a locally AMUC space which has either a normalized, symmetric basic sequence which is not equivalent to the unit vector basis in $\ell_1$, or a normalized sequence with upper $p$-estimates for some $1<p<\infty$. Then $Y$ fails property B.

Theorems & Definitions (11)

  • Theorem A
  • Proposition 1: Corollary 2.3 DKRRZ
  • Lemma 2
  • Proposition 3
  • proof
  • Proposition 4: Proposition $4$ Aguirre
  • Theorem 5
  • Proposition 6
  • proof
  • Theorem 7
  • ...and 1 more