Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions

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. The main result of this talk 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, using probabilistic methods and an old result of Rev. Simmons on flipping biased coins. Time permitting we will consider 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 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 d-by-d symmetric matrices in a free spectrahedron D_L dilates, up to a scale factor, to a tuple T of commuting self-adjoint operators with joint spectrum in the spectrahedron S_L. From another viewpoint, the scale factor measures the extent that a positive map can fail to be completely positive.