Some permanence results of the Dunford-Pettis and Grothendieck properties in lcHs
Grothendieck spaces with the Dunford-Pettis property play an important role in the theory of Banach spaces and vector measures. These two properties have been investigated recently in the context of locally convex spaces and sequence spaces, mainly in connection with mean ergodic operators on Fréchet spaces. See, e.g.,\ the article by \textit{W. Ricker} and the reviewer in [Positivity 11, No.~1, 77--93 (2007; Zbl 1131.46005)]. The aim of the article under review is to study other permanence properties of the Dunford-Pettis and the Grothendieck properties for non-normable locally convex spaces. Section 2 investigates permanence results under the formation of separated quotients. As a consequence it is proved that if the Köthe echelon space \(\lambda_0(A)\) of order zero is quasinormable, then the quotient \(\lambda_\infty(A)/\lambda_0(A)\) is a Grothendieck space with the Dunford-Pettis property. Several results on (LF)-spaces of sequences are included. The last section is devoted to show several permanence results under the formation of projective and injective tensor products of Fréchet spaces. This question is related to the problem of topologies of Grothendieck for projective tensor products, and to work by Domański, Lindström and Schlüchtermann in the 1990's for injective tensor products.
scientific article; zbMATH DE number 5914710
scientific article