MULTIPLICATION OPERATORS ON VECTOR-VALUED FUNCTION SPACES


Duru H. , KITOVER A., Orhon M.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, vol.141, pp.3501-3513, 2013 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 141 Issue: 10
  • Publication Date: 2013
  • Doi Number: 10.1090/s0002-9939-2013-11603-5
  • Title of Journal : PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
  • Page Numbers: pp.3501-3513

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 Kothe-Bochner space. An operator on E(X) is called a multiplication operator if it is given by multiplication by a function in L-infinity (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-infinity (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, Rodriguez, and Sanchez-Perez.