Classes of Dunford-Pettis-type operators with applications to Banach spaces and Banach lattices
Zeekoei, Elroy Denovanne
MetadataShow full item record
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 , leads to the introduction of the concept of weak∗ p-convergent operator in this study (and in the paper ). 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 ) 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 . 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
Showing items related by title, author, creator and subject.
A study of Dunford–Pettis–like properties with applications to polynomials and analytic functions on normed spaces Zeekoei, Elroy Denovanne (North-West University, 2011)Recall that a Banach space X has the Dunford-Pettis property if every weakly compact operator defined on X takes weakly compact sets into norm compact sets. Some valuable characterisations of Banach spaces with the ...
Zeekoei, Elroy D.; Fourie, Jan H. (Springer, 2018)The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sánchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present ...
Fourie, Jan H.; Zeekoei, Elroy D. (Taylor & Francis, 2017)The purpose of this article is to introduce and study the notion of “weak* p-convergent operator”. We discuss the relationship between the weak* p-convergent operators and the p-convergent operators, a class of operators ...