Extension of Pettis integration: Pettis operators and their integrals
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In this note, the authors discuss the concepts of a Pettis operator, by which they mean a weak\(^*\)–weakly continuous linear operator F from a dual Banach space to an \(L_1\)-space, and of its Pettis integral, understood simply as the dual operator \(F^*\) of F. Applications to radial limits in weak Hardy spaces of vector-valued harmonic and holomorphic functions are provided.
KeywordsPettis integral Pettis operator Weak\(^*\)–weakly continuous operator Weak Hardy space
Mathematics Subject Classification46G10 28B05
The first author has been supported by grant MTM2014-53009-P (MINECO, Spain). Both the authors are grateful to A. Michalak, K. Musiał, M. Nawrocki and W. Ruess for their interest in this work and numerous helpful comments. We also thank the referee for his/her interesting comments which helped us to improve the paper.
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