Positive operator-valued noncommutative polynomials are squares

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 $$p=r^{*}r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml: