On approximation properties of $l_{1}$ -type 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 ∞ 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.