Vector and operator valued holomorphic functions representable by Carleman type formulas

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Abstract

Let be a simply connected domain and let M be a connected subset of its boundary of positive Lebesque measure. With X we denote a separable Hilbert space or the space of bounded linear functionals on . We set f to be an X-valued holomorphic function, and with we denote the class of X-valued holomorphic functions on which belong to the Hardy class near the set M. In our main result, we show that if f belongs to Near Hardy Class, then f is representable by a Carleman type formula, and conversely, if f is representable by a Carleman type formula, and in some sense has an analytic continuation across M, then f belongs to the above Near Hardy class . Furthermore we show that in general Near hardy and classical Hardy classes do not coincide.
Original languageEnglish
Pages (from-to)1117-1128
Number of pages12
JournalComplex Variables, Theory and Applications
Volume49
Issue number15
Publication statusPublished - 2004

Keywords

  • Hardy classes
  • Carleman formulas
  • Operator valued Holomorphic Functions

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