On the convergence of a sequence of positive linear operators on the space of M-multiple complex sequences

Abstract

Let $Lambda _p$ be the space of m-multiple complex numerical sequence $alpha = (alpha_k), k=(k_1,...,k_m), k_j geq 0, 1 leq j leq m $, such that $|alpha_k|leq M_{alpha p(k)}$, where p(k) = $1+ ||k||^2 = 1 + k_1^2 +.. .+k_m^2$ and $M_{alpha}$ is a positive constant depending on $alpha = (alpha_k)$. We denote by $Delta^+_p$ the set of $(alpha_k)in Lambda_p$, all of whose components are non-negative. In this paper, we investigate the convergence problem for a sequence of linear operators $T_n$ acting from $Lambda_p$ to $Lambda_p$ and satisfying the condition $T_nLambda_p^+subsetlambda_p^+$.