Introducing Statistical Operators: Boundedness, Continuity, and Compactness

TL;DR

This paper introduces the concepts of statistical boundedness, continuity, and compactness for operators between normed spaces, establishing their relationships with classical operator theory and exploring their properties.

Contribution

It is the first to systematically study statistical operator properties, bridging statistical convergence with traditional operator theory in normed spaces.

Findings

Defined statistical boundedness, continuity, and compactness for operators.

Established connections between statistical and classical operator concepts.

Provided examples illustrating statistical operator behavior.

Abstract

Many studies have been conducted on statistical convergence, and it remains an area of active research. Since its introduction, statistical convergence has found applications many fields. Nevertheless, there is a shortage of research related to operator theory. As far as we know, no studies have focused on continuous, bounded, and compact operators, which are fundamental concepts in mathematics. We explore the notions of statistical boundedness, continuity, and compactness of operators between normed spaces, establishing connections between these concepts and their counterparts in traditional normed space theory. Additionally, we provide examples and results that demonstrate the behavior and implications of statistical convergence in the context of operators.