On convergence with respect to an ideal and a family of matrices

TL;DR

This paper generalizes recent convergence concepts involving matrices and Orlicz functions by replacing single matrices and functions with families, and applies these ideas to Banach space theory, extending Simons' theorem.

Contribution

It introduces a generalized framework for convergence using families of matrices and Orlicz functions, expanding previous notions and providing new applications in Banach space theory.

Findings

Generalization of $A^I$-summability and $A^I$-statistical convergence.

Extension of Simons' $ ext{sup}$-$ ext{limsup}$-theorem to new convergence methods.

Application to Banach space theory with convergence methods involving countable bases.

Abstract

Recently P. Das, S. Dutta and E. Savas introduced and studied the notions of strong $A^I$-summability with respect to an Orlicz function $F$ and $A^I$-statistical convergence, where $A$ is a non-negative regular matrix and $I$ is an ideal on the set of natural numbers. In this note, we will generalise these notions by replacing $A$ with a family of matrices and $F$ with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' $\sup$-$\limsup$-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal $I$ has a countable base), continuing the author's previous work.