1. Introduction One branch of operator theory which has blossomed during the last four decades is the study of Hilbert spaces operators related to the unit disk D = { z G C : |Z| 1 } . Although its origin can be traced to the von Neumann-Wold characterization of isometry [28, 32], the subject began in earnest with Beurl- ing's determination of the invariant subspaces of the shift operator [10] and von Neumann's work on spectral sets [29]. In [24] B.Sz.-Nagy gave an alternative proof of von Neumann's theorem that the unit disk is a spectral set for contrac- tions via unitary dilations and then developed the latter notion in collaboration with C. Foias into a model theory for contraction operators [27]. Operators of class Co with spectrum in the closed unit disk were introduced by B.Sz.-Nagy and C. Foias in their work on canonical models for contractions. A completely nonunitary contraction belongs to this class if the associated functional calculus on H°° has a non-trivial kernel. Since when the class Co was first defined [24], it was studied by many others (see [7]) and, quite possibly, is the best understood class of non-normal operators. In this paper O will be a bounded finitely connected region in the complex plane, whose boundary T consists of disjoint, analytic, simple closed curves. Let R(Q) be the space of rational functions with poles off ^, and Rat(f2) be the closure in C(£2) of R(Q). Let H be a complex Hilbert space and let C(H) be the algebra of bounded linear operators on H. M.B. Abrahamse and R.G. Douglas [4] initiated in 1974 the study of contractive unital £(H^representation of Rat(fi). In this paper and two years later in a paper about subnormal operators related to multiply connected regions [5], they quoted a preprint in preparation about Co operators over multiply connected regions, that was never published. In 1978 in a paper about operators of class Coo over multiply connected regions [6] J.A. Ball refers to operators of class Co for finitely connected regions saying without proof that an operator of class Co is of class Coo too. Abrahamse and Douglas were planning to develop the theory of the class Co starting from their study of bundle shifts [5]. Our approach here avoids almost completely the use of bundle shifts. Bundle shifts would be necessary in a theory where unitary equivalence is the basic classification criterium, while we only consider similarity and quasisimilarity. We consider linear bounded operators on a Hilbert space having Q as spec- tral set, and no normal summand with spectrum in I\ For each operator satisfying these properties, we define a weak*-continuous functional calculus $ : H°°(Q,) — • C(H), where if°°(fi) is the Banach algebra of bounded analytic functions on 17 (under somewhat more restrictive hypotheses this functional cal- culus was considered earlier by B. Chevreau, CM. Pearcy and A.L. Shields [11]). An operator is said to be of class Co if the associated functional calculus has a non-trivial kernel. The central object of this paper are operators of class Co, for which we provide a complete classification into quasisimilarity classes analogous to Jordan's classical results in finite-dimensional linear algebra. Our work can be viewed as an extension to more general regions of results first proved for the l

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