Multiplication operators on vector-valued function spaces
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- by Hülya Duru, Arkady Kitover and Mehmet Orhon PDF
- Proc. Amer. Math. Soc. 141 (2013), 3501-3513 Request permission
Abstract:
Let $E$ be a Banach function space on a probability measure space $(\Omega ,\Sigma ,\mu ).$ Let $X$ be a Banach space and $E(X)$ be the associated Köthe-Bochner space. An operator on $E(X)$ is called a multiplication operator if it is given by multiplication by a function in $L^{\infty }(\mu ).$ In the main result of this paper, we show that an operator $T$ on $E(X)$ is a multiplication operator if and only if $T$ commutes with $L^{\infty }(\mu )$ and leaves invariant the cyclic subspaces generated by the constant vector-valued functions in $E(X).$ As a corollary we show that this is equivalent to $T$ satisfying a functional equation considered by Calabuig, Rodríguez, and Sánchez-Pérez.References
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Additional Information
- Hülya Duru
- Affiliation: Department of Mathematics, Faculty of Science, Istanbul University, Vezneciler- Istanbul, 34134, Turkey
- Email: hduru@istanbul.edu.tr
- Arkady Kitover
- Affiliation: Department of Mathematics, Community College of Philadelphia, 1700 Spring Garden Street, Philadelphia, Pennsylvania 19130
- Email: akitover@ccp.edu
- Mehmet Orhon
- Affiliation: Department of Mathematics and Statistics, University of New Hampshire, Durham, New Hampshire 03824
- Email: mo@unh.edu
- Received by editor(s): April 5, 2011
- Received by editor(s) in revised form: December 15, 2011
- Published electronically: June 17, 2013
- Additional Notes: The first author was supported by the Scientific Projects Coordination Unit of Istanbul University, project No. 3952
- Communicated by: Thomas Schlumprecht
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3501-3513
- MSC (2010): Primary 47B38; Secondary 46G10, 46B42, 46H25
- DOI: https://doi.org/10.1090/S0002-9939-2013-11603-5
- MathSciNet review: 3080172