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
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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 $0\leq A_0\in L^{r}_{n}(Y, Z)$, $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ 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