Journal of Operator Theory

Volume 41, Issue 2, Spring 1999 pp. 321-350.

Model theory for $\rho$-contractions, $\rho \leq 2$

Authors: Michael A. Dritschel (1), Scott McCullough (2), and Hugo J. Woerdeman (3)
Author institution: (1) Department of Mathematics, Purdue University, West Lafayette, Indiana 47097-1395, USA
(2) Department of Mathematics, The University of Florida, Gainsville, Florida 32611-2082, USA
(3) Department of Mathematics, The College of William & Mary Williamsburg, Virginia 23185-8795, USA

Summary: 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 $\Crho$.

Keywords: Model theory, $\rho$-contractions, numerical radius, extension, factor ization, extremal, complete positivity, Schur complement, operator-valuate d analytic function, convexity

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