  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.
