Dilations of matricies

We explore aspects of dilation theory in the finite dimensional case and show that for a commuting $n$-tuple of operators $T=(T_1,...,T_n)$ acting on some finite dimensional Hilbert space $H$ and a compact set $X\subset \mathbb{C}^n$ the following are equivalent: 1. $T$ has a normal $X$-dilation. 2. For any $m\in \mathbb{N}$ there exists some finite dimensional Hilbert space $K$ containing $H$ and a tuple of commuting normal operators $N=(N_1,...,N_n)$ acting on $K$ such that $q(T)=P_Hq(N)|_H$ for all polynomials $q$ of degree at most $m$ and such that the joint spectrum of $N$ is contained in $X$ (where $P_H$ is the projection from $K$ to $H$).

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