Factoring Non-negative Operator Valued Trigonometric Polynomials in Two Variables

Abstract

It is shown using Schur complement techniques that on finite dimensional Hilbert spaces, a non-negative operator valued trigonometric polynomial in two variables with degree $(d_1,d_2)$ can be written as a finite sum of hermitian squares of at most $2d_2$ analytic polynomials.

20 pages. The main result holds only over finite dimensional Hilbert spaces and the minor corrections needed for this are incorporated towards the end of Section 3. The version of record of this article, first published in Mathematische Annalen, is available online at the publisher's website: https://doi.org/10.1007/s00208-024-02895-9