2012/08/17 by Sriram Balasubramanian, Scott McCullough, Balasubramanian, Sriram +1
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #math.FA
paper · pdf · doi:10.48550/arxiv.1208.3582
arxiv created 2012/08/17 · openalex publication_date 2012/08/17 · arxiv updated 2012/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \Rx denote the ring of polynomials in g freely non-commuting variables x=(x1,...,xg). There is a natural involution * on \Rx determined by xj^*=xj and (pq)^*=q^* p^* and a free polynomial p∈\Rx is symmetric if it is invariant under this involution. If X=(X1,...,Xg) is a g tuple of symmetric n× n matrices, then the evaluation p(X) is naturally defined and further p^*(X)=p(X)^*. In particular, if p is symmetric, then p(X)^*=p(X). The main result of this article says if p is symmetric, p(0)=0 and for each n and each symmetric positive definite n× n matrix A the set X:A-p(X)\succ 0 is convex, then p has degree at most two and is itself convex, or -p is a hermitian sum of squares.