Sequences of linear operators related to Cesàro-convergent sequences

Abstract

Given a Cesàro-convergent sequence of real numbers$(a_n{)}_{n\in \mathbb{N}}$, a sequence $({\phi }_n{)}_{n\in \mathbb{N}}$ of operators is defined on the Banach space $\mathcal{R}(I,F)$ of regular functions defined on $I=[0,1]$ and having values in a Banach space $F$, $${\phi }_n(f)=\frac{1}{n}\sum _{k=1}^n{a}_{k}f\left(\frac{k}{n}\right).$$It is proved that if, in addition, the sequence $(\frac{\left|a_1\right|+\dots +\left|a_n\right|}{n}{)}_{n\in \mathbb{N}}$ is bounded, then ${\phi }_n(f)$ converges to $a\cdot {\int }_0^{1}f,$ where $a=\underset{n\to \mathrm{\infty }}{lim}\frac{a_1+\dots +a_n}{n}.$ The converse of this statement is also true. Another result is that the supplementary condition can be dropped if the operators are considered on the space ${\mathcal{C}}^1(I,F)$