Classes of Dunford-Pettis-type operators with applications to Banach spaces and Banach lattices

Abstract

The aim of this study is to extend existing knowledge of Dunford-Pettis operators on Banach spaces and Banach lattices and their variants. Using the concept of p-convergent operators as basis, we introduce the Dunford- Pettis-type operators by introducing the so-called weak p-convergent, disjoint p-convergent as well as the weak∗ p-convergent operators on Banach spaces and Banach lattices. The concept of the DP∗Pp on Banach spaces, which was introduced in the paper [37], leads to the introduction of the concept of weak∗ p-convergent operator in this study (and in the paper [67]). An interesting fact is that for the Banach spaces, most of the existing results in the study field of Dunford-Pettis operators and their variants may be carried over in our context where we consider the weak p-summable sequences and base most of our definitions and results on these types of sequences. However, the Banach lattice cases are a bit trickier and we have to revert to crafty plans in order to find proofs for our results. We also study the coincidences of these types of operators as well as a type of domination property that each of these types of operators possess. A classic property is that of the Schur property, which is used to characterise the almost Dunford-Pettis operators. Associated with this is the so-called positive Schur property of Banach lattices. We follow these concepts and develop the so-called Schur property of order p as well as its positive version. A discussion of sequentially p-limited operators introduced by Karn and Sinha (see [48]) to study the p-DPP, follows. Motivated by the Banach ideal property of (Ltp, ℓtp), we introduce the general concept of “operator [Y, p]-summable sequence in a Banach space X, consider the vector space Yp(X) of all operator [Y, p]-summable sequences in X and then introduce a norm on the space. Yp(X) turns out to be a Banach space and we apply the results of the general setting to the special setting of operator p-summable sequences in a Banach space X. This leads to the extension and improvement of results in [48]. We then apply the concept of a disjoint p-convergent operator to introduce the so-called disjoint p-convergent functions on Banach lattices. Our specific focus here is to establish under what conditions continuous n-homogeneous polynomials, holomorphic maps and symmetric separately compact bilinear maps are disjoint p-convergent functions