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Abstract  

Defant [5] introduced the local Radon–Nikodým property for duals of locally convex spaces. This is a generalization of Asplund spaces as defined in Banach space theory. In this paper we generalise Dunford"s Theorem [7] to Banach spaces with Schauder decompositions and apply this result to spaces of holomorphic functions on balanced domains in a Banach space.

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We prove that some results on uniform convergence of sequences of unconditionally convergent series, in Banach spaces, can be generalized to sequences of weakly unconditionally Cauchy series.

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Abstract  

Some atomic decomposition theorems are proved in vector-valued weak martingale Hardy spaces w pΣα(X), w p Q α(X) and wD α(X). As applications of atomic decompositions, a sufficient condition for sublinear operators defined on some vector-valued weak martingale Hardy spaces to be bounded is given. In particular, some weak versions of martingale inequalities for the operators f*, S (p)(f) and σ(p)(f) are obtained.

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We establish the existence of mild solutions and periodic mild solutions for a class of abstract first-order non-autonomous neutral functional differential equations with infinite delay in a Banach space.

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In this paper atomic decompositions for two-parameter vector-valued martingales are given. With the help of the atomic decompositions the relations between the mutual embedding of two-parameter vector-valued martingale spaces and geometric properties of Banach spaces are investigated. Our study shows that geometric properties of Banach spaces determine the embedding of martingale spaces and conversely the latter can characterize the former.

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Abstract  

A pair of linear bounded commuting operators T1, T2 in a Banach space is said to possess a decomposition property (DePr) if Ker (I-T1)(I-T2) = Ker (I-T1) + Ker (I-T2). A Banach space X is said to possess a 2-decomposition property (2-DePr) if every pair of linear power bounded commuting operators in X possesses the DePr. It is known from papers of M. Laczkovich and Sz. Rvsz that every reflexive Banach space X has the 2-DePr. In this paper we prove that every quasi-reflexive Banach space of order 1 has the 2-DePr but not all quasi-reflexive spaces of order 2. We prove that a Banach space has no 2-DePr if it contains a direct sum of two non-reflexive Banach spaces. Also we prove that if a bounded pointwise norm continuous operator group acts on X then every pair of operators belonging to it has a DePr. A list of open problems is also included.

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