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 $V:\mathcal H\to \mathcal K$ 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 $d\times d$ symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space.