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Two-sided multiplication operators on the space of regular operators


Authors: Jin Xi Chen and Anton R. Schep
Journal: Proc. Amer. Math. Soc. 144 (2016), 2495-2501
MSC (2010): Primary 46A40; Secondary 46B42, 47B65
DOI: https://doi.org/10.1090/proc/12893
Published electronically: October 21, 2015
MathSciNet review: 3477065
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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)$, $ \vert M_{A_0, B}\vert(T)=M_{A_0, \vert B\vert}(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 $ \vert M_{A,\, B}\vert=M_{\vert A\vert, \,\vert B\vert}$ 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.


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Additional 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
Email: schep@math.sc.edu

DOI: https://doi.org/10.1090/proc/12893
Keywords: Regular operator, two-sided multiplication operator, Riesz space, Banach lattice
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
Article copyright: © Copyright 2015 American Mathematical Society

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