Archimedean operator-theoretic Positivstellens\"atze

TL;DR

This paper establishes a broad archimedean Positivstellensatz for hermitian operator-valued polynomials, unifying and extending several classical results in operator theory and polynomial positivity.

Contribution

It introduces a general archimedean Positivstellensatz for hermitian operator-valued polynomials, generalizing multiple existing theorems in the field.

Findings

Unified framework for Positivstellensatz in operator-valued polynomials

Generalizations of Fejer-Riesz and related theorems

Extension of archimedean Positivstellensatz for *-algebras

Abstract

We prove a general archimedean positivstellensatz for hermitian operator-valued polynomials and show that it implies the multivariate Fejer-Riesz Theorem of Dritschel-Rovnyak and positivstellens\"atze of Ambrozie-Vasilescu and Scherer-Hol. We also obtain several generalizations of these and related results. The proof of the main result depends on an extension of the abstract archimedean positivstellensatz for *-algebras that is interesting in its own right.