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

TL;DR

This paper characterizes classes of holomorphic functions with positive real part on the unit ball using operator-valued Herglotz formulas, explores duality relations, and identifies extreme points within these classes.

Contribution

It introduces a family of weighted pairings on the ball, establishes duality relations, and identifies self-dual classes and extreme points in the context of positive Schur functions.

Findings

Defined operator-valued Herglotz formulas for classes of functions

Established duality relations between classes of functions

Identified extreme points of the positive Schur class

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-Fantappi\`e 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.