Type of Publication:  Preprint 
Author:  Helton, J. William; Klep, Igor; McCullough, Scott A.; Schweighofer, Markus 
Year of publication:  2015 
ArXivID:  arXiv:1412.1481v2 
Summary: 
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 dbyd symmetric matrices of operator norm at most one to a collection of commuting selfadjoint contraction operators on a Hilbert space. An analytic formula for ϑ(d) is derived, which as a byproduct 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=(A_1,...,A_g) of symmetric matrices of the same size, L(x):=I\sum A_j x_j is a monic linear pencil. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is a spectrahedron. The set D_L of tuples X=(X_1,...,X_g) of symmetric matrices (of the same size) for which L(X):=I\sum A_j \otimes X_j is PsD, is a free spectrahedron. A result here is: any tuple X of dbyd symmetric matrices in a bounded free spectrahedron D_L dilates, up to a scale factor, to a tuple T of commuting selfadjoint operators with joint spectrum in the corresponding spectrahedron S_L. 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 D_L \subset D_M obviously implies the inclusion S_L \subset S_M 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 D_L gives a precise measure of the worst case error inherent in the free relaxation, over all monic linear pencils M of size d.

Subject (DDC):  510 Mathematics 
Keywords:  Functional Analysis, Optimization and Control, Probability 
Bibliography of Konstanz:  Yes 
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HELTON, J. William, Igor KLEP, Scott A. MCCULLOUGH, Markus SCHWEIGHOFER, 2015. Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions
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