Let \(L\) be a monic linear pencil in \(g\) variables (i.e., there is a natural number \(l\) and symmetric \(l\times l\) matrices \(A_1,\ldots,A_g\) such that \(L=I_l-\sum_{j=1}^gA_jx_j\) where \(x_1,\ldots,x_g\) are noncommuting variables). The monic linear pencil induces a family of semialgebraic sets \(\mathfrak{P}_L=(\mathfrak{P}_L(n))_{n\in\mathbb{N}}\) where \[ \mathfrak{P}_L(n):=\{X\in \mathbb{S}_n^g\mid L(X)\succeq 0\}, \] and \(\mathbb{S}_n^g\) is the set of \(g\)-tuples of symmetric \(n\times n\) matrices. By the conditions imposed on \(L\) each \(\mathfrak{P}_L(n)\) is convex with interior. Conversely by J. W. Helton and S. McCullough [Ann. Math. (2) 176, No. 2, 979–1013 (2012; Zbl 1260.14011)], convex bounded noncommutative semialgebraic sets with interior are all of the form \(\mathfrak{P}_L\).
The main result of the paper under review establishes a perfect noncommutative Nichtnegativstellensatz on \(\mathfrak{P}_L\). It states that a symmetric matrix-valued noncommutative polynomial \(p\) is positive semidefinite on \(\mathfrak{P}_L\) (i.e. \(p(X)\succeq 0\) for all \(X\in \mathfrak{P}_L(n)\) and all \(n\)) if and only if \(p\) has a weighted sum of squares representation with optimal degree bounds: \[ p=s^*s+\sum_j^{\mathrm{finite}}f_j^*Lf_j \] where \(s,f_j\) are matrices of noncommutative polynomials of degree no greater than \(\mathrm{deg}(p)/2\) (and \(*\) is the natural involution on matrix-valued noncommutative polynomials which maps \(x_ix_j\) to \(x_jx_i\)).
A similar result is also shown in the more general setting that the underlying semialgebraic set is defined by a concave matrix-valued noncommutative polynomial.
The Nichtnegativstellensatz in the cases described above improves strongly the result of J. W. Helton and S. McCullough [Trans. Am. Math. Soc. 356, No. 9, 3721–3737 (2004; Zbl 1071.47005)] where the commutative Positivstellensatz of M. Putinar [Indiana Univ. Math. J. 42, No. 3, 969–984 (1993; Zbl 0796.12002)] has been adapted to the noncommutative setting.