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.