Some permanence results of the Dunford-Pettis and Grothendieck properties in lcHs
type: Item
instance of: scholarly article
title: Some permanence results of the Dunford-Pettis and Grothendieck properties in lcHs
zbMATH Open document ID: 1226.46002
author: Angela A. Albanese
author: Elisabetta M. Mangino
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2011-06-29T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1308749127
review text: 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.
reviewed by: José Bonet
Mathematics Subject Classification ID: 46A03
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A20
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 5914710
zbMATH Keywords: locally convex spaces, Fréchet spaces
zbMATH Keywords: Grothendieck spaces
zbMATH Keywords: Dunford-Pettis property
zbMATH Keywords: Schur property
zbMATH Keywords: tensor products
MaRDI profile type: Publication
OpenAlex ID: W2043045821
cites work: Grothendieck spaces with the Dunford-Pettis property
cites work: \(C_{0}\)-semigroups and mean ergodic operators in a class of Fréchet spaces
cites work: A question of D. Vogt on (LF)-spaces
cites work: Q4652139
cites work: On the Dunford-Pettis property of the tensor product of \(C(K)\) spaces
cites work: The Problem of Topologies of Grothendieck and the Class of Fréchet T-Spaces
cites work: On the lifting of bounded sets in Fréchet spaces
cites work: Tensor stable Fréchet and (DF)-spaces
cites work: Schauder decompositions and the Grothendieck and Dunford--Pettis properties in Köthe echelon spaces of infinite order
cites work: The Schur property on projective and injective tensor products
cites work: New classes of \(L_ p-\)spaces
cites work: Q3688898
cites work: Montel Subspaces in the Countable Projective Limits of Lp(μ)-Spaces
cites work: On quotients of Köthe sequence spaces of infinite order
cites work: Distinguished Fréchet spaces and projective tensor product
cites work: On Fréchet Montel spaces and their projective tensor product
cites work: Grothendieck Spaces and Duals of Injective Tensor Products
cites work: Grothendieck operators on tensor products
cites work: Remarks on Weak Compactness of Operators Defined on Certain Injective Tensor Products
cites work: Q4892384
cites work: Q5488192
cites work: Grothendieck locally convex spaces of continuous vector valued functions
cites work: The Dunford-Pettis property on tensor products
cites work: Q3918541
cites work: A Note on Frechet-Montel Spaces
cites work: A Note on Banach Spaces Containing ℓ1
cites work: Q3803669
cites work: La propriété de Dunford-Pettis dans \(C(K,E)\) et \(L^ 1(E)\)
cites work: Q3752948
cites work: Q4015685
DOI: 10.7169/FACM/1308749127
Recommended article: Grothendieck spaces with the Dunford-Pettis property
Recommended article: Schauder decompositions and the Grothendieck and Dunford--Pettis properties in Köthe echelon spaces of infinite order
Recommended article: Dunford-Pettis type properties and the Grothendieck property for function spaces
Recommended article: Q4338527
Recommended article: On the projective tensor product of Fréchet spaces
Recommended article: Q4282711
Recommended article: Q3752948
Recommended article: Q3695846
Recommended article: Q4872736
description: scientific article; zbMATH DE number 5914710