Positive operator-valued noncommutative polynomials are squares (2511.06487v1)

Abstract: 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 key technical input is a canonical tuple of self-adjoint operators $A$ and unitaries $U$, respectively, 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 $S$ representing the sos polynomial $p$. As a consequence, the cone of (sums of) squares of polynomials is closed in the weak operator topology. Exploiting this closedness, the GNS construction associates to a separating linear functional a finite-rank positive semidefinite noncommutative Hankel matrix and, on its range, produces the desired tuple $Y$.