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