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 Caratheodory's and of Minkowski's theorems for matrix convex sets.

Dilation theory in finite dimensions and matrix convexity

Lupini M
2021

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 Caratheodory's and of Minkowski's theorems for matrix convex sets.
File in questo prodotto:
File Dimensione Formato  
fd_dilation-10.pdf

accesso aperto

Tipo: Postprint
Licenza: Licenza per accesso libero gratuito
Dimensione 666.36 kB
Formato Adobe PDF
666.36 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/914629
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 5
  • ???jsp.display-item.citation.isi??? 4
social impact