On pseudo weakly compact operators of order $ P$

TL;DR

This paper introduces and studies pseudo weakly compact operators of order p, explores related properties like p-Dunford-Pettis, and characterizes Banach spaces with these properties, including tensor product conditions.

Contribution

It defines new classes of operators and properties in Banach space theory, providing characterizations and conditions for these properties to hold.

Findings

Introduction of pseudo weakly compact operators of order p.

Characterizations of Banach spaces with p-Dunford-Pettis property.

Conditions for tensor products to have p-sequentially Right properties.

Abstract

In this paper, we introduce the concept of a pseudo weakly compact operator of order $ p $ between Banach spaces. Also we study the notion of $ p $-Dunford-Pettis relatively compact property which is in "general" weaker than the Dunford-Pettis relatively compact property and gives some characterizations of Banach spaces which have this property. Moreover, by using the notion of $ p $-Right subsets of a dual Banach space, we study the concepts of $ p $-sequentially Right and weak $ p $-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spaces $ X$ and $ Y $ such that projective tensor and injective tensor products between $ X $ and $ Y $ have the $ p $-sequentially Right property.\ Finally, we introduce two properties for the Banach spaces, namely $ p $-sequentially Right$ ^{\ast} $ and weak $ p $-sequentially Right$ ^{\ast} $ properties and obtain some characterizations of these properties.