Operator-valued Herglotz kernels and functions of positive real part on the ball

2008/01/02 by Michael T. Jury, Jury, Michael T. · 1 citation
Mathematics · #Holomorphic and Operator Theory #Analytic and geometric function theory #Algebraic and Geometric Analysis

paper · pdf · doi:10.48550/arxiv.0801.0438

Abstract

We describe several classes of holomorphic functions of positive real part on the unit ball; each is characterized by an operator-valued Herglotz formula. Motivated by results of J.E. McCarthy and M. Putinar, we define a family of weighted Cauchy-Fantappiè pairings on the ball and establish duality relations between certain pairs of classes, and in particular we identify the dual of the positive Schur class. We also establish the existence of self-dual classes with respect to this pairing, and identify some extreme points of the positive Schur class.

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