Positive operator-valued noncommutative polynomials are squares

Jindal, Abhay (author), Klep, Igor (author), McCullough, Scott (author)

We establish operator-valued versions of the earlier foundational factorization results for noncommutative polynomials due to Helton (Ann. Math., 2002) and one of the authors (Linear Alg. Appl., 2001). Specifically, we show that every positive operator-valued noncommutative polynomial $p$ admits a single-square factorization $p=r^* r$. An analogous statement holds for operator-valued noncommutative trigonometric polynomials (i.e., operator-valued elements of a free group algebra). Our approach follows the now standard sum-of-squares (sos) paradigm but requires new results and constructions tailored to operator coefficients. Assuming a positive $p$ is not sos, Hahn-Banach separation yields a linear functional that is positive on the sos cone and negative on $p$; a Gelfand-Naimark-Segal (GNS) construction then produces a representing tuple $Y$ leading to contradiction since $p$ was assumed positive on $Y$. The main technical input is a canonical tuple $A$ of self-adjoint operators and, in the unitary case, a canonical tuple $U$ of unitaries, both constructed from the left-regular representation on Fock space. We prove that, up to a universal constant, the norms $\|p(A)\|$ and $\|p(U)\|$ bound the operator norm of any positive semidefinite Gram matrix $G$ representing the sos polynomial $p$. This uniform control is the key input in showing that the cone of (sums of) squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then produces a separating functional that is continuous for the weak operator topology (WOT). This two-step passage between the ultraweak and WOT topologies constitutes our separation argument and yields the required WOT closedness of the sos cone. With this in hand, the GNS construction associates to such a separating linear functional a finite-rank positive semidefinite noncommutative Hankel matrix and, on its range, produces the desired tuple $Y$.

• Language: English
• Keywords: factorization, noncommutative polynomial, sum of squares, trigonometric polynomial, Positivstellensatz
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.03.2026
• Year: 2026
• Number of pages: 26 str.
• Numbering: Vol. 98, iss. 1, article no. 7
• PID: 20.500.12556/RUL-180864
• UDC: 517.9
• ISSN on article: 0378-620X
• DOI: 10.1007/s00020-026-02828-y
• COBISS.SI-ID: 272097539
• Publication date in RUL: 18.03.2026

Record is a part of a journal

• Title: Integral equations and operator theory
• Shortened title: Integr. equ. oper. theory
• Publisher: Springer
• ISSN: 0378-620X
• COBISS.SI-ID: 25623552

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• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.