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 ttd, then it admits a certificate of weighted degree at most ttd+cS, where cS 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 ttd independently of K, we show that no uniform additive constant exists for natural matrix preorderings. We construct quadratic 2×2 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 ttd+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.
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BibTeX
@misc{jindal2026operator,
title = {{Operator Fejér--Riesz Theorems with Gaps and Degree Bounds}},
author = {Abhay Jindal and Aljaž Zalar},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02967},
eprint = {2611.02967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02967},
}
RIS
TY - UNPB
TI - Operator Fejér--Riesz Theorems with Gaps and Degree Bounds
AU - Abhay Jindal
AU - Aljaž Zalar
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02967
UR - https://arxiv.org/abs/2611.02967
AB - 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 ttd, then it admits a certificate of weighted degree at most ttd+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 ttd independently of K, we show that no uniform additive constant exists for natural matrix preorderings. We construct quadratic 2×2 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 ttd+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.
ER -