### Model theory for $\rho$-contractions, $\rho\le2$ (1999)

Author(s): Dritschel MA, McCullough S, Woerdeman HJ

Abstract: Agler's abstract model theory is applied to $\Crho$, the family of operators with unitary $\rho$-dilations, where $\rho$ is a fixed number in $(0,2]$. The extremals, which are the collection of operators in $\Crho$ with the property that the only extensions of them which remain in the family are direct sums, are characterized in a variety of manners. They form a part of any model, and in particular, of the boundary, which is defined as the smallest model for the family. Any model for a family is required to be closed under direct sums, restrictions to reducing subspaces, and unital $*$-representations. In the case of the family $\Crho$ with $\rho\in(0,1)\cup(1,2]$, this closure is shown to be all of $\Crho$.

• Date: 1999
• Journal: Journal of Operator Theory
• Volume: 41
• Issue: 2
• Pages: 321-350
• Publisher: The Theta Foundation
• Publication type: Article
• Bibliographic status: Published

Keywords: Model theory, p-contractions, numerical radius, extension, factorization, extremal, complete positivity, Schur complement, operator-valuated analytic function, convexity

 Dr Michael Dritschel Reader in Pure Mathematics Email: michael.dritschel@ncl.ac.uk Telephone: +44 (0) 191 208 7229