Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions
J. William Helton, Igor Klep, Scott McCullough, Markus Schweighofer · Memoirs of the American Mathematical Society · 2019
An operator C C on a Hilbert space H \mathcal H dilates to an operator T T on a Hilbert space K \mathcal K if there is an isometry V : H → K V:\mathcal H\to \mathcal K such that C = V ∗ T V C= V^* TV . A main result of this paper is, for a positive integer d d , the simultaneous dilation, up to a sharp factor ϑ ( d ) \vartheta (d) , expressed as a ratio of Γ \Gamma functions for d d even, of all d × d d\times d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space. Dilating to commuting operators has consequences for the theory of linear matrix inequalities (LMIs). Given a tuple A = ( A 1 , … , A g ) A=(A_1,\dots ,A_g) of ν