Caratheodory-Fejer interpolation and related topics in locally convex spaces

TL;DR

This paper investigates Caratheodory-Herglotz functions valued in operator spaces over locally convex spaces, reducing the problem to bounded operators on Hilbert spaces, thus extending classical interpolation theory.

Contribution

It introduces a reduction technique for Caratheodory-Herglotz functions in locally convex spaces to bounded operators on Hilbert spaces, broadening the scope of existing interpolation methods.

Findings

Reduction of operator-valued functions to Hilbert space cases

Extension of Caratheodory-Fejer interpolation theory

Application to locally convex topological spaces

Abstract

We study Caratheodory-Herglotz functions whose values are continuous operators from a locally convex topological space which admits the factorization property into its conjugate dual space. We show how this case can be reduced to the case of functions whose values are bounded operators from a Hilbert space into itself.