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## Model theory for $\rho$-contractions, $\rho\le2$

Lookup NU author(s): Dr Michael DritschelORCiD

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### 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 $\Crho$.

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

Publication type: Article

Publication status: Published

Journal: Journal of Operator Theory

Year: 1999

Volume: 41

Issue: 2

Pages: 321-350

Print publication date: 01/01/1999

ISSN (print): 0379-4024

Publisher: The Theta Foundation