Schauder decompositions and the Grothendieck and Dunford--Pettis properties in Köthe echelon spaces of infinite order
A Fréchet space \(X\) is called a Grothendieck space if every sequence in the dual space \(X'\) which is convergent for the weak* topology \(\sigma(X',X)\) is also convergent for the weak topology \(\sigma(X',X'')\). A Fréchet space \(X\) is said to have the Dunford--Pettis property if every element of \(L(X,Y)\), for \(Y\) an arbitrary quasicomplete locally convex Hausdorff space (\(L(X,Y)\) denotes the space of all linear continuous maps from \(X\) into \(Y\)), which transforms bounded subsets of \(X\) into relatively weakly compact subsets of \(Y\), also transforms weakly compact subsets of \(X\) into compact subsets of \(Y\). The class of Fréchet spaces which are Grothendieck spaces with the Dunford--Pettis property are denoted by GDP-spaces. Well-known examples of GDP-spaces include \(L^\infty\), \(H^\infty ( D)\), injective Banach spaces, and certain \(C(K)\) spaces. In the setting of non-normable Fréchet spaces, the only known examples of GDP-spaces were given by Fréchet--Montel spaces. Other than Montel spaces, the authors show that every Köthe echelon space \(\lambda_\infty(A)\), with \(A\) an arbitrary Köthe matrix, is a GDP-space. In the case when \(\lambda_\infty(A)\not =\lambda_0(A)\) (note that \(\lambda_\infty(A) =\lambda_0(A)\) if and only if \(\lambda_\infty(A)\) is Montel), the authors show that \(\lambda_\infty(A)\) still admits unconditional Schauder decompositions, provided that it satisfies the density condition. This contrasts the situation for GDP-Banach spaces which do not have any Schauder decomposition by a result of \textit{D.\,Dean} [Proc.\ Am.\ Math.\ Soc.\ 18, 619--623 (1967; Zbl 0158.13503)]. Consequences for spectral measures are also given.
scientific article; zbMATH DE number 5156944