2014/12/31 by J. William Helton, Igor Klep, Scott A. McCullough +1 · 2 citations
Mathematics · #math.FA #math.OC #math.PR #msc:47A20 #msc:46L07 #msc:13J30 #msc:60E05 #msc:33B15 #msc:90C22
published as Mem. Amer. Math. Soc. 257 vol. 1232 (2019), 104pp · v4: 93 pages including a table of contents and index, added paragraph at the end of the introduction, several minor changes; v3: 90 pages, new section on including balls into spectrahedra, changes to the introduction; v2: title change, introduction rewritten; v1: 77 pages including a table of contents
arxiv created 2016/05/07 · arxiv updated 2019/06/05
An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V from H to K such that C=V^*TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor ϑ(d), of all d-by-d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space. An analytic formula for ϑ(d) is derived, which as a by-product gives new probabilistic results for the binomial and beta distributions. Dilating to commuting operators has consequences for the theory of linear matrix inequalities (LMIs). Given a tuple A=(A1,...,Ag) of symmetric matrices of the same size, L(x):=I-∑ Aj xj is a monic linear pencil. The solution set SL of the corresponding linear matrix inequality, consisting of those x in Rg for which L(x) is positive semidefinite (PsD), is a spectrahedron. The set DL of tuples X=(X1,...,Xg) of symmetric matrices (of the same size) for which L(X):=I-∑ Aj ⊗ Xj is PsD, is a free spectrahedron. A result here is: any tuple X of d-by-d symmetric matrices in a bounded free spectrahedron DL dilates, up to a scale factor, to a tuple T of commuting self-adjoint operators with joint spectrum in the corresponding spectrahedron SL. From another viewpoint, the scale factor measures the extent that a positive map can fail to be completely positive. Given another monic linear pencil M, the inclusion DL ⊂ DM obviously implies the inclusion SL ⊂ SM and thus can be thought of as its free relaxation. Determining if one free spectrahedron contains another can be done by solving an explicit LMI and is thus computationally tractable. The scale factor for commutative dilation of DL gives a precise measure of the worst case error inherent in the free relaxation, over all monic linear pencils M of size d.