Cancellability and Regularity of Operator Connections

TL;DR

This paper introduces and characterizes the properties of cancellability and regularity in operator connections, exploring their mathematical structure and solution behavior for related operator equations.

Contribution

It provides new definitions, characterizations, and analysis of cancellability and regularity in operator connections, including existence and uniqueness results for operator equations.

Findings

Characterization of cancellability and regularity in operator connections

Existence and uniqueness results for operator equations involving these connections

Insights into the structure of operator monotone functions and Borel measures

Abstract

An operator connection is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above and the transformer inequality. In this paper, we introduce and characterize the concepts of cancellability and regularity of operator connections with respect to operator monotone functions, Borel measures and certain operator equations. In addition, we investigate the existence and the uniqueness of solutions for such operator equations.