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C. Carathéodory und L. Fejér, “über den Zusammenhang der Extreme von Harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz,” Rendiconti del Circolo Matematico di Palermo, Vol. 32, No. 1, 1911, pp. 218-239. doi:10.1007/BF03014796

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• TITLE: Positive-Definite Operator-Valued Kernels and Integral Representations

  AUTHORS: L. Lemnete-Ninulescu

  KEYWORDS: Unitary-Operator; Self-Adjoint Operator; Joint Spectral Measure of a Commuting Tuple of Operators; Spectral Projector; Complex Moments; Analytic Vectorial Functions

  JOURNAL NAME: Applied Mathematics, Vol.3 No.12, December 24, 2012

  ABSTRACT: A truncated trigonometric, operator-valued moment problem in section 3 of this note is solved. Let be a finite sequence of bounded operators, with arbitrary, acting on a finite dimensional Hilbert space H. A necessary and sufficient condition on the positivity of an operator kernel for the existence of an atomic, positive, operator-valued measure , with the property that for every with , the moment of coincides with the term of the sequence, is given. The connection between some positive definite operator-valued kernels and the Riesz-Herglotz integral representation of the analytic on the unit disc, operator-valued functions with positive real part in the class of operators in Section 4 of the note is studied.