Disjoint p $p$ -convergent operators and their adjoints

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 $T:E\to F$ is almost weak $p$-convergent if and only if $T^{\ast }$ is disjoint $p$-convergent; (ii) $E^{\ast }$ has order continuous norm or $F^{\ast }$ has the positive Schur property of order $p$. Very recent results are improved, examples are given and applications of the main results are provided.