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Multiplication operators on vector-valued function spaces

Authors: Hülya Duru, Arkady Kitover and Mehmet Orhon
Journal: Proc. Amer. Math. Soc. 141 (2013), 3501-3513
MSC (2010): Primary 47B38; Secondary 46G10, 46B42, 46H25
Published electronically: June 17, 2013
MathSciNet review: 3080172
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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.

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Additional Information

Hülya Duru
Affiliation: Department of Mathematics, Faculty of Science, Istanbul University, Vezneciler- Istanbul, 34134, Turkey

Arkady Kitover
Affiliation: Department of Mathematics, Community College of Philadelphia, 1700 Spring Garden Street, Philadelphia, Pennsylvania 19130

Mehmet Orhon
Affiliation: Department of Mathematics and Statistics, University of New Hampshire, Durham, New Hampshire 03824

Keywords: Multiplication operator, K\"othe-Bochner space, vector-valued measurable function, Banach function space, Banach lattice, ideal center, Banach $C(K)$-module
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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