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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
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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  • [1] Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298 (2006h:46005)
  • [2] Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Springer, Dordrecht, 2006. Reprint of the 1985 original. MR 2262133
  • [3] Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, 2nd ed., Mathematical Surveys and Monographs, vol. 105, American Mathematical Society, Providence, RI, 2003. MR 2011364 (2005b:46010)
  • [4] Pere Ara and Martin Mathieu, Local multipliers of $ C^*$-algebras, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2003. MR 1940428 (2004b:46071)
  • [5] Raúl E. Curto, Spectral theory of elementary operators, Elementary operators and applications (Blaubeuren, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 3-52. MR 1183936 (93i:47041)
  • [6] Elementary operators and their applications, Operator Theory: Advances and Applications, vol. 212, Birkhäuser/Springer Basel AG, Basel, 2011. Papers from the 3rd International Workshop held at Queen's University Belfast, Belfast, April 14-17, 2009; Edited by Raúl E. Curto and Martin Mathieu. MR 2789128 (2012e:47003)
  • [7] Martin Mathieu, The norm problem for elementary operators, Recent progress in functional analysis (Valencia, 2000) North-Holland Math. Stud., vol. 189, North-Holland, Amsterdam, 2001, pp. 363-368. MR 1861772 (2002g:47071),
  • [8] Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093 (93f:46025)
  • [9] Eero Saksman and Hans-Olav Tylli, Multiplications and elementary operators in the Banach space setting, Methods in Banach space theory, London Math. Soc. Lecture Note Ser., vol. 337, Cambridge Univ. Press, Cambridge, 2006, pp. 253-292. MR 2326390 (2008i:47076),
  • [10] J. Synnatzschke, Über einige verbandstheoretische Eigenschaften der Multiplikation von Operatoren in Vektorverbänden, Math. Nachr. 95 (1980), 273-292 (German). MR 592901 (82b:47048),
  • [11] A. W. Wickstead, Norms of basic elementary operators on algebras of regular operators, Proc. Amer. Math. Soc., electronically published on May 22, 2015,
  • [12] Adriaan C. Zaanen, Introduction to operator theory in Riesz spaces, Springer-Verlag, Berlin, 1997. MR 1631533 (2000c:47074)

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Additional Information

Jin Xi Chen
Affiliation: Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, People’s Republic of China

Anton R. Schep
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

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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