Riesz multiplier convergent spaces of operator valued series and a version of Orlicz- Pettis theorem

TL;DR

This paper introduces Riesz multiplier convergent spaces for operator valued series, establishes their completeness criteria, and presents a new version of the Orlicz-Pettis theorem using Riesz summability, advancing the understanding of operator series convergence.

Contribution

It develops a generalized Riesz multiplier convergence framework for operator series and links it to completeness and classical theorems like Orlicz-Pettis.

Findings

Defined the space $M^ Infty_{R}$ of Riesz summability for operator series.

Provided completeness criteria for these spaces with $c_0(X)$-multiplier convergence.

Established a new version of the Orlicz-Pettis theorem using Riesz summability.

Abstract

It is not usual to characterize an operator valued series via completeness of multiplier spaces. In this study, by using a series of bounded linear operators, we introduce the space $M^\infty_{R}\big(\sum_k T_k\big)$ of Riesz summability which is a generalization of the Ces\`{a}ro summability. Therefore, we give the completeness criteria of these spaces with $c_0(X)$-multiplier convergent operator series. It is a natural consequence that one can characterize the completeness of a normed space through $M^\infty_{R}\big(\sum_k T_k\big)$ which will be assumed that is complete for every $c_0(X)$-multiplier Cauchy operator series. Then, we characterize the continuity and the (weakly) compactness of the summing operator $\mathcal{S}$ from the multiplier space $M^\infty_{R}\big(\sum_k T_k\big)$ to an arbitrary normed space $Y$ through $c_0(X)$-multiplier Cauchy and $\ell_\infty(X)$-multiplier convergent series, respectively. We also prove that if $\sum_kT_k$ is $\ell_\infty(X)$-multiplier Cauchy, then the multiplier space of weakly Riesz-convergence associated to the operator valued series $M^\infty_{wR}\big(\sum_k T_k\big)$ is subspace of $M^\infty_{R}\big(\sum_k T_k\big)$. Among other results, finally, we obtain a new version of the well-known Orlicz-Pettis theorem by using Riesz-summability.