Substochastic operators in symmetric spaces

First, we solve a crucial problem under which conditions increasing uniform K$K$‐monotonicity is equivalent to lower local uniform K$K$‐monotonicity. Next, we investigate properties of substochastic operators on L1+L∞$L^1+L^\infty$ with applications. Namely, we show that a countable infinite combination of substochastic operators is also substochastic. Using K$K$‐monotonicity properties, we prove several theorems devoted to the convergence of the sequence of substochastic operators in the norm of a symmetric space E$E$ under addition assumption on E$E$. In our final discussion, we focus on compactness of admissible operators for Banach couples under additional assumption.