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Representation of sequence classes by operator ideals

Geraldo Botelho, Ariel S. Santiago

Abstract

It is well known that weakly $p$-summable sequences in a Banach space $E$ are associated to bounded operators from $\ell_{p^*}$ to $E$, and unconditionally $p$-summable sequences in $E$ are associated to compact operators from $\ell_{p^*}$ to $E$. Generalizing these results to a quite wide environment, we characterize the classes of Banach spaces-valued sequences that are associated to (or represented by) some Banach operator ideal. Using these characterizations, we decide, among all sequence classes that usually appear in the literature, which are represented by some Banach operator ideal and which are not. Moreover, to each class that is represented by some Banach operator ideal, we show an ideal that represents it. Illustrative examples and additional applications are provided.

Representation of sequence classes by operator ideals

Abstract

It is well known that weakly -summable sequences in a Banach space are associated to bounded operators from to , and unconditionally -summable sequences in are associated to compact operators from to . Generalizing these results to a quite wide environment, we characterize the classes of Banach spaces-valued sequences that are associated to (or represented by) some Banach operator ideal. Using these characterizations, we decide, among all sequence classes that usually appear in the literature, which are represented by some Banach operator ideal and which are not. Moreover, to each class that is represented by some Banach operator ideal, we show an ideal that represents it. Illustrative examples and additional applications are provided.
Paper Structure (4 sections, 13 theorems, 54 equations)

This paper contains 4 sections, 13 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ideal-representable sequence classes
  4. Further applications

Key Result

Proposition 3.3

Let $\lambda$ be a scalar sequence space. (a) $\lambda_*$ is a linear space with the usual algebraic operations, the map is a norm on $\lambda_*$ and the correspondence is an isometric isomorphism. In particular, $\lambda_*$ is a Banach space. (b) For every $(\alpha_j)_{j=1}^\infty\in \lambda$, $(\alpha_j)_{j=1}^\infty=\sum\limits_{j=1}^\infty \alpha_j e_j$. (c) $c_{00} \subseteq \lambda_* \stac

Theorems & Definitions (34)

  • Example 2.1
  • Definition 3.1
  • Proposition 3.3
  • Definition 3.4
  • Remark 3.5
  • Example 3.6
  • Proposition 3.7
  • proof
  • Example 3.8
  • Lemma 3.9
  • ...and 24 more