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

An operator $C$ on a Hilbert space $mathcal H$ dilates to an operator $T$ on a Hilbert space $mathcal K$ if there is an isometry $V:mathcal Hto mathcal K$ 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 $vartheta (d)$, expressed as a ratio of $Gamma $ functions for $d$ even, of all $dtimes d$ symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space.