Substochastic operators in symmetric spaces

TL;DR

This paper investigates properties of substochastic operators in symmetric spaces, establishing conditions for K-monotonicity equivalences, their stability under infinite combinations, and convergence behaviors, with applications to compactness in Banach couples.

Contribution

It provides new conditions linking different types of K-monotonicity, analyzes the stability of substochastic operators under infinite sums, and explores their convergence and compactness properties.

Findings

Countable infinite sums of substochastic operators are substochastic.

Established equivalence conditions for increasing uniform and lower locally uniform K-monotonicity.

Proved convergence of substochastic operator sequences in symmetric spaces.

Abstract

First, we solve a crucial problem under which conditions increasing uniform K-monotonicity is equivalent to lower locally uniform K-monotonicity. Next, we investigate properties of substochastic operators on $L^1+L^\infty$ with applications. Namely, we show that a countable infinite combination of substochastic operators is also substochastic. Using K-monotonicity properties, we prove several theorems devoted to the convergence of the sequence of substochastic operators in the norm of a symmetric space E under addition assumption on E. In our final discussion we focus on compactness of admissible operators for arbitrary Banach couples.