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On limited and almost limited operators between Banach lattices

Safak Alpay, Svetlana Gorokhova

TL;DR

This work analyzes limited and almost limited operators between Banach lattices, clarifying how dual lattice continuity, order-interval structure, and norm o-continuity influence inclusions among DP, Lwc, semi-compact, and related operator classes. It establishes precise conditions under which semi-K, DP, W, Lwc, Mwc, and their almost variants collapse to more compact or limited classes, including when $E^*$ is a KB-space or when $F$ has limited/compact order intervals or o-continuous norms. It extends the theory to regular/positive and factorized operators, and develops collective versions for families of operators, providing lattice-approximation tools for collectively Mwc families. The results unify and extend known inclusions, yielding practical criteria for when operators between Banach lattices exhibit compactness-type behavior in both individual and collective settings.

Abstract

We study (almost) limited operators in Banach lattices and their relations to L-weakly compact, semi-compact, and Dunford-Pettis operators. Several further related topics are investigated.

On limited and almost limited operators between Banach lattices

TL;DR

This work analyzes limited and almost limited operators between Banach lattices, clarifying how dual lattice continuity, order-interval structure, and norm o-continuity influence inclusions among DP, Lwc, semi-compact, and related operator classes. It establishes precise conditions under which semi-K, DP, W, Lwc, Mwc, and their almost variants collapse to more compact or limited classes, including when is a KB-space or when has limited/compact order intervals or o-continuous norms. It extends the theory to regular/positive and factorized operators, and develops collective versions for families of operators, providing lattice-approximation tools for collectively Mwc families. The results unify and extend known inclusions, yielding practical criteria for when operators between Banach lattices exhibit compactness-type behavior in both individual and collective settings.

Abstract

We study (almost) limited operators in Banach lattices and their relations to L-weakly compact, semi-compact, and Dunford-Pettis operators. Several further related topics are investigated.
Paper Structure (4 sections, 37 theorems, 21 equations)

This paper contains 4 sections, 37 theorems, 21 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. The effect of w$^\ast$-continuity of lattice operations in the dual
  3. Almost limited operators
  4. Collectively limited and almost limited operator families

Key Result

Lemma 1.1

(AlBu and Mey91) TFAE.

Theorems & Definitions (78)

  • Lemma 1.1
  • Definition 1.2
  • Lemma 1.3
  • Definition 1.4
  • Definition 1.5
  • proof
  • Example 1.8
  • Definition 1.9
  • Definition 1.10
  • Lemma 2.1
  • ...and 68 more