On Cesàro summability of vector valued multiplier spaces and operator valued series
The authors establish analogues for operator-valued series of some results on completeness of normed spaces due to \textit{A. Aizpuru} and \textit{F. J. Pérez-Fernández} [Acta Math. Univ. Comen., New Ser. 68, No.~2, 337--344 (1999; Zbl 0952.46009)] using Cesàro convergence. Let \(X,Y\) be normed spaces and \(L(X,Y)\) the space of continuous linear operators from \(X\) into \(Y\). A (formal) series \(\sum_{k}x_{k}\) in \(X\) is Cesàro convergent to \(x_{0}\) if \(C-\sum_{k}x_{k}=\lim_{n}\frac{1}{n}\sum_{k=1}^{n}x_{k}=x_{0}\). If \(F\) is a vector space of \(X\)-valued sequences, a series \(\sum_{k}T_{k}\) in \(L(X,Y)\) is \(F\)-multiplier Cesàro convergent (Cauchy) if the series \(\sum_{k}T_{k}x_{k}\) is Cesàro convergent (Cauchy) for every \(\{x_{k}\}\in F\). For a series \(\sum_{k}T_{k}\) in \(L(X,Y)\), define \[ M_{C}^{\infty }(\{T_{k}\})=\left\{x=\{x_{k}\}\in l^{\infty }(X):\sum_{k}T_{k}x_{k}\text{ is Cesàro convergent}\right\} \] and equip this space with the sup-norm. Define the summing operator \(S:M_{C}^{\infty }(\{T_{k}\})\rightarrow Y\) by \(Sx=C-\sum_{k=1}^{\infty }T_{k}x_{k}\). The following results are typical of what is established: {\parindent=6mm \begin{itemize}\item[(1)] If \(X,Y\) are Banach spaces, then \(M_{C}^{\infty }(\{T_{k}\})\) is complete iff \(\sum_{k}T_{k}\) is \(c_{0}(X)\) multiplier convergent iff \(\sum_{k}T_{k}x_{k}\) is Cesàro convergent for every \(\{x_{k}\}\in c_{0}(X)\). \item[(2)] The summing operator \(S\) is continuous iff \(\sum_{k}T_{k}\) is \(c_{0}(X)\)-multiplier Cauchy. \end{itemize}} Other similar results concerning completeness and multiplier convergent series are established. The authors also introduce the space \(M_{wC}^{\infty }(\{T_{k}\})\) of weakly Cesàro multiplier convergent series of operators and study similar results. A version of the Orlicz-Pettis theorem for Cesàro convergent series of operators is established.
scientific article; zbMATH DE number 6873077
scientific article