Operator Fejér--Riesz Theorems with Gaps and Degree Bounds

Abstract

Let $K$ be a nonempty compact semialgebraic subset of the real line, and let $S$ be a saturated scalar description of $K$. We prove that every hermitian polynomial with bounded-operator coefficients that is positive semidefinite on $K$ belongs to the operator quadratic module generated by $S$. We also show that if such a polynomial has degree at most $\tt{d}$, then it admits a certificate of weighted degree at most $\tt{d}+c_S$, where $c_S$ depends only on $S$, not on the polynomial or the coefficient Hilbert space. In contrast to the scalar case, where natural preorderings admit the degree bound $\tt{d}$ independently of $K$, we show that no uniform additive constant exists for natural matrix preorderings. We construct quadratic $2\times2$ matrix polynomials positive semidefinite on unions of two intervals whose minimum certificate degrees in the corresponding natural preorderings diverge as one interval shrinks to a point, while the gap length remains bounded below. Conversely, we establish uniform bounds of the form $\tt{d}+C$ for families of interval unions contained in a fixed compact interval when interval lengths and gap lengths have uniform positive lower bounds. Our proofs develop two techniques that may be of independent interest. First, a quadratic reduction principle represents arbitrary-degree operator polynomials as compressions of quadratic ones on enlarged Hilbert spaces. A perturbation argument and the closedness of truncated operator quadratic modules then extend uniform degree bounds from the quadratic case to arbitrary degrees with linear degree bounds of slope $1$. Second, a rank-one symmetric matrix pencil transfers positivity on interval unions to $[0,1]$, where the operator Fejér--Riesz theorem yields certificates.