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.
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