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

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Hardy Class
Analytic function
Operator
Denote
Separable Hilbert Space
Linear Functionals
Analytic Continuation
Subset

Keywords

  • Hardy classes
  • Carleman formulas
  • Operator valued Holomorphic Functions

Cite this

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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.",
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pages = "1117--1128",
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Vector and operator valued holomorphic functions representable by Carleman type formulas. / Chailos, George.

In: Complex Variables, Theory and Applications, Vol. 49, No. 15, 2004, p. 1117-1128.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Vector and operator valued holomorphic functions representable by Carleman type formulas

AU - Chailos, George

PY - 2004

Y1 - 2004

N2 - 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.

AB - 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.

KW - Hardy classes

KW - Carleman formulas

KW - Operator valued Holomorphic Functions

M3 - Article

VL - 49

SP - 1117

EP - 1128

JO - Complex Variables, Theory and Applications

JF - Complex Variables, Theory and Applications

SN - 1747-6941

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