×

Sequential convergences and Dunford-Pettis properties. (English) Zbl 0969.46031

Let \(X\) and \(Y\) be Banach spaces, \({\mathcal A}(X,Y)\) a class of functions \(X \to Y\) and \(\tau\) a sequential convergence on \(X\). We say that \({\mathcal A}\) has the \(\tau\)-Dunford-Pettis property on \(X\) if for each Banach space \(Y\), each weakly compact function \(F \in {\mathcal A}(X,Y)\) and each \(\tau\)-null sequence \((x_n)\) in \(X\) we have \(F(x_n) \to F(0)\) in the norm topology on \(Y\). That \(F\) is weakly compact means that it maps some neighborhood of the origin in \(X\) into a weakly relatively compact subset of \(Y\). The goal of the paper under review is to give a uniform approach to \(\tau\)-Dunford-Pettis properties. It is shown that for any \(\tau\) one obtains the same Dunford-Pettis property for \({\mathcal A}\) the class of linear operators, \(k\)-homogeneous polynomials or holomorphic functions which are bounded on bounded subsets of \(Y\). The situation changes if \({\mathcal A}\) is the class of all bounded holomorphic functions. Finally it is shown that if \(\tau\) is the holomorphic convergence, i.e., \(x_n \to_\tau x\) if and only if for all holomorphic functions \(f \: X \to \mathbb C\) we have \(f(x_n) \to f(x)\), then for each Banach space \(X\) even the strongest Dunford-Pettis property (\({\mathcal A}\) is the class of all holomorphic functions) holds.

MSC:

46G20 Infinite-dimensional holomorphy
46B20 Geometry and structure of normed linear spaces
PDFBibTeX XMLCite
Full Text: EuDML EMIS