Introducing Statistical Operators: Boundedness, Continuity, and Compactness

Abstract

Many studies have been conducted on statistical convergence, which remains an area of active research. Since its introduction, statistical convergence has found applications in many fields. Nevertheless, there is a shortage of studies related to operator theory; especially of those on continuous, bounded, and compact operators. We explore the notions of statistical boundedness, continuity, and compactness of operators between normed spaces, establishing connections between these concepts and their counterparts in the traditional normed space theory. Additionally, we provide some examples and results that demonstrate the behavior and implications of statistical convergence in operator theory.