Substochastic operators in symmetric spaces

First, we solve a crucial problem under which conditions increasing uniform $$K$$-monotonicity is equivalent to lower local 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 Banach couples under additional assumption.