Statistical order convergence of operators on Riesz Spaces

TL;DR

This paper introduces statistical order convergence for sequences of operators on Riesz spaces, establishing its properties, relationship to classical convergence, and demonstrating its weaker nature through examples, thus extending operator convergence concepts.

Contribution

It defines statistical order convergence for operators, proves its key properties, and links it to classical order convergence, expanding the framework of unbounded operator convergence.

Findings

Statistical order convergence is weaker than classical order convergence.

Fundamental properties like uniqueness and stability are established.

Examples illustrate the proper extension of convergence notions.

Abstract

This paper introduces statistical order convergence and its pointwise variant for sequences of order bounded operators between Riesz spaces. We establish fundamental properties: uniqueness of the limit, stability under lattice operations, and a characterization via natural density linking it to classical order convergence. Explicit examples show that statistical order convergence is strictly weaker than order convergence, confirming that this concept provides a proper extension of operator-theoretic convergence notions. The results preserve essential lattice structures and open avenues for further research in unbounded convergence and Banach lattice theory.