Disjoint $p$-convergent operators and their adjoints

TL;DR

This paper characterizes when disjoint $p$-convergent operators coincide with almost Dunford-Pettis operators in Banach lattices and explores the conditions under which adjoints of positive operators are disjoint $p$-convergent.

Contribution

It provides new conditions on Banach lattices for equivalence of disjoint $p$-convergence and almost Dunford-Pettis properties, and characterizes when adjoints of positive operators are disjoint $p$-convergent.

Findings

Conditions on Banach lattices for operator equivalences

Equivalence of operator properties with order continuity or positive Schur property

Improved recent results with new examples and applications

Abstract

First we give conditions on a Banach lattice $E$ so that an operator $T$ from $E$ to any Banach space is disjoint $p$-convergent if and only if $T$ is almost Dunford-Pettis. Then we study when adjoints of positive operators between Banach lattices are disjoint $p$-convergent. For instance, we prove that the following conditions are equivalent for all Banach lattices $E$ and $F$: (i) A positive operator $T \colon E \to F$ is almost weak $p$-convergent if and only if $T^*$ is disjoint $p$-convergent; (ii) $E^*$ has order continuous norm or $F^*$ has the positive Schur property of order $p$. Very recent results are improved, examples are given and applications of the main results are provided.