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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Two-sided multiplication operators on the space of regular operators
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by Jin Xi Chen and Anton R. Schep
Proc. Amer. Math. Soc. 144 (2016), 2495-2501
DOI: https://doi.org/10.1090/proc/12893
Published electronically: October 21, 2015

Abstract:

Let $W$, $X$, $Y$ and $Z$ be Dedekind complete Riesz spaces. For $A\in L^{r}(Y, Z)$ and $B\in L^{r}(W, X)$ let $M_{A, B}$ be the two-sided multiplication operator from $L^{r}(X, Y)$ into $L^r(W, Z)$ defined by $M_{A, B}(T)=ATB$. We show that for every $0\leq A_0\in L^{r}_{n}(Y, Z)$, $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ holds for all $B\in L^{r}(W, X)$ and all $T\in L^{r}_{n}(X, Y)$. Furthermore, if $W$, $X$, $Y$ and $Z$ are Dedekind complete Banach lattices such that $X$ and $Y$ have order continuous norms, then $|M_{A, B}|=M_{|A|, |B|}$ for all $A\in L^{r}(Y, Z)$ and all $B\in L^{r}(W, X)$. Our results generalize the related results of Synnatzschke and Wickstead, respectively.
References
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Bibliographic Information
  • Jin Xi Chen
  • Affiliation: Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, People’s Republic of China
  • Email: jinxichen@home.swjtu.edu.cn
  • Anton R. Schep
  • Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
  • MR Author ID: 155835
  • Email: schep@math.sc.edu
  • Received by editor(s): October 30, 2014
  • Received by editor(s) in revised form: July 8, 2015
  • Published electronically: October 21, 2015
  • Additional Notes: The first author was supported in part by China Scholarship Council (CSC) and was visiting the University of South Carolina when this work was completed.
  • Communicated by: Thomas Schlumprecht
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 2495-2501
  • MSC (2010): Primary 46A40; Secondary 46B42, 47B65
  • DOI: https://doi.org/10.1090/proc/12893
  • MathSciNet review: 3477065