Herglotz representation for operator-valued function on a set associated with test functions

TL;DR

This paper extends the classical Herglotz representation theorem to operator-valued functions on arbitrary sets using modern realization techniques and test functions, broadening its applicability in operator theory.

Contribution

It introduces a novel reinterpretation of the Herglotz representation for operator-valued functions via realization formulas and test functions, applicable to arbitrary sets.

Findings

Extended Herglotz representation to operator-valued functions

Connected classical theory with modern realization techniques

Applicable to functions on arbitrary sets

Abstract

The Herglotz representation theorem for holomorphic functions with non-negative real part is a fundamental result in the theory of holomorphic functions. In this paper, we reinterpret the Herglotz representation in the context of modern techniques, specifically realization formula. This reinterpretation is then extended to operator-valued functions on arbitrary sets, in association with a collection of test functions.