The Grothendieck Space Property for Tensor Products of Banach Spaces
by
Qingying Bu
The University of Mississippi
Coauthors: Donghai Ji and Xiaoping Xue

In this paper we obtain the following results: (a) If both X and Y are Grothendieck spaces and each continuous linear operator from X to Y* is compact then the projective tensor product of X and Y is a Grothendieck space. (b) Suppose that either X* or Y* has the Radon-Nikodym property and that either X** or Y** has the approximation property. If both X and Y are Grothendieck spaces and each continuous linear operator from X* to Y** is compact then the injective tensor product of X and Y is a Grothendieck space. (c) The conditions for the projective (resp. injective) tensor product of X and Y being a Grothendieck space in (a) (resp. in (b)) are not only sufficient but also necessary in case that X has an unconditional finite dimensional decomposition.

Date received: January 17, 2008