Convergence properties of nets of operators

Abstract

We consider nets $\left(T_j\right)$ of operators acting on complex functions, and we investigate the algebraic and the topological structure of the set $\{f:T_j(\left|f{|}^2\right)-|T_{j}f{|}^2\to 0\}$. Our results extend and improve some known results from the literature, which are connected with Korovkin’s theorem. Applications to Abel–Poisson-type operators and Bernstein-type operators are given.