Pontryagin Space Structure in Reproducing Kernel Hilbert Spaces over *-semigroups

TL;DR

This paper explores the connection between Pontryagin space structures and *-semigroups through positive definite functions and reproducing kernel Hilbert spaces, advancing the understanding of indefinite inner product spaces in operator theory.

Contribution

It establishes a novel link between Pontryagin space geometry and *-semigroup algebra via reproducing kernel Hilbert spaces and positive definite functions.

Findings

Characterization of elements determining shift operators as Pontryagin symmetries

Link between indefinite inner product spaces and algebraic structures of *-semigroups

Development of conditions for shift operators in Krein spaces

Abstract

The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of *-semigroups. It goes via the positive definite functions and related to them reproducing kernel Hilbert spaces. Our concern is in describing properties of elements of the semigroup which determine shift operators which serve as Pontryagin fundamental symmetries