Convergence of sequences of k-positive linear operators in subspaces of the space of analytic functions

Abstract

Let A be the space of analytic functions on the unit disc and fk, k = 0, 1, 2, . . ., the Taylor coefficients of a function $fin A$. For any real number r, define a subspace Ar of A as follows Ar = ${fin A:|fk|leq M_f(1 + k^r)}$.Let $T_n | A rightarrow A$ be a sequence of linear k-positive operators in the sense of [3]and $phi_ s(z)= sumlimits_{k=0}^infty k^s z^k$The purpose of this study is to find a number s such that $parallel T_n f(z)-f(z)parallel Arightarrow 0$ for every $fin A$ as soon as $parallel T_nphi_{sv}parallel A$ tends to zero for v = 0, 1, 2.