Model Theory for - Contractions , 2

Agler's abstract model theory is applied to C, the family of operators with unitary-dilations, where is a xed number in (0; 2]. The extremals, which are the collection of operators in C 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 deened 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 C with 2 (0; 1) (1; 2], this closure is shown to be all of C. Throughout this paper C , a xed positive constant, stands for the class of Hilbert space operators with unitary-dilations; that is, a bounded linear operator A acting on a Hilbert space H (notation: A 2 L(H)) belongs to C if and only if there is a Hilbert space K containing H isometrically and a unitary operator U 2 L(K) such that (0.1) where P H is the orthogonal projection from K onto H. The elements of C are referred to as-contractions. The-contractions were introduced by Holbrook 12] and Sz.-Nagy and Foia s 15] as a natural generalization of the usual contractions (which correspond to the case where = 1). In this paper, we consider model theory for the class C when 0 < 2, extending the work of the rst and third authors in 9] on the class C 2 , the operators with numerical radius less than or equal to one. In general, the point of model theory is to study some class of operators through a smaller, better understood collection of operators which have the property that the elements of the original class are the restriction to invariant subspaces of elements of the smaller collection. This is what we mean by \modeling". It is a concept which has been widely applied in operator theory. In order to better understand what is going on in a host of seemingly unrelated examples, Jim Agler codiied in 2] an abstract model theory. According to Agler, model theory is applied to a \family" of operators (formal deenitions are given below). There are potentially