A study of reciprocal Dunford-Pettis-like properties on Banach spaces

TL;DR

This paper explores reciprocal Dunford-Pettis-like properties in Banach spaces, focusing on dual space subsets, operator properties, and tensor product stability, advancing understanding of operator compactness and space characterization.

Contribution

It introduces the q-reciprocal Dunford-Pettis* property of order p and characterizes Banach spaces with this property, also analyzing its stability under tensor products.

Findings

Characterization of Banach spaces with q-reciprocal Dunford-Pettis* property

Relationship between p-(V) and p-V* subsets in dual spaces

Stability results for the property under projective tensor products

Abstract

In this article, we study the relationship between \(p\)-\((V)\) subsets and p-\(V^*\) subsets of dual spaces. We investigate the Banach space X with the property that adjoint every \(p\)-convergent operator \(T: X \rightarrow Y\) is weakly \(q\)-compact, for every Banach space \(Y\). Moreover, we define the notion of \(q\)-reciprocal Dunford-Pettis\(\^*\)property of order \(p\) on Banach spaces and obtain a characterization of Banach spaces with this property. The stability of reciprocal Dunford-Pettis property of order \(p\) for the projective tensor product is given.