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

Understand

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 $\vartheta(d)$, 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 $\vartheta(d)$ is derived, which as a by-product gives new probabilistic results for the binomial and beta distributions.
• Dilating to commuting operators has consequences for the theory of linear matrix inequalities (LMIs).