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arXiv · 1910.03549

Dilation theory in finite dimensions and matrix convexity

Michael Hartz·Martino Lupini
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Abstract

We establish a finite-dimensional version of the Arveson-Stinespring dilation theorem for unital completely positive maps on operator systems. This result can be seen as a general principle to deduce finite-dimensional dilation theorems from their classical infinite-dimensional counterparts. In addition to providing unified proofs of known finite-dimensional dilation theorems, we establish finite-dimensional versions of Agler's theorem on rational dilation on an annulus, of Berger's dilation theorem for operators of numerical radius at most $1$, and of the Putinar-Sandberg numerical range dilation theorem. As a key tool, we prove versions of Carathéodory's and of Minkowski's theorem for matrix convex sets.

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Michael Hartz, Martino Lupini. 2020-09-15. Dilation theory in finite dimensions and matrix convexity. https://doi.org/10.1007/s11856-021-2202-5

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