Characterization of $M$-compact sets via statistically convergent sequences

TL;DR

This paper investigates the stability and characterization of $M$-compact sets in Banach spaces, introducing statistically convergent sequences and $ ext{I}$-$M$-compactness, expanding understanding of compactness notions in functional analysis.

Contribution

It provides a new characterization of $M$-compact sets via statistically maximizing sequences and introduces $ ext{I}$-$M$-compactness, linking it to existing $M$-compactness concepts.

Findings

Stability results for $M$-compactness in $l^p$ sums of Banach spaces.

Characterization of $M$-compact sets using statistically maximizing sequences.

Equivalence of $ ext{I}$-$M$-compactness and $M$-compactness for non-trivial admissible ideals.

Abstract

In this paper, we study stability of $M$-compactness for $l^p$ sum of Banach spaces for $1\leq p<\infty$. We also obtain a characterization of $M$-compact sets in terms of statistically maximizing sequence, a notion which is weaker than a maximizing sequence. Moreover, we introduce the notion of $\mathcal{I}$-$M$-compactness of a bounded subset $M$ of a normed linear space $X$ with respect to an ideal $\mathcal{I}$ and show that it is equivalent to $M$-compactness for non-trivial admissible ideals.