OPERATORS OF CLASS C0 3

general planar regions. Lastly we discuss certain properties of H°°(Q) that will

be essential in what follows. First we explain the generalization of the concepts

of divisibility, greatest common inner divisor and least common inner multiple

to functions over multiply connected regions; then we deal with some analytical

properties of inner functions in order to define the concept of support of inner

functions, in analogy with the same concept for inner functions on the disk.

The third chapter contains the theory of operators of class Co. We intro-

duce the class and study its elementary properties related to adjoints, invariant

subspaces and functional calculus. We show that operators of class Co have a

minimal inner function, which is analogous to the minimal polynomial of a finite

matrix, and that every inner function occurs as the minimal function of some

operator of class Co- Then we define operators of class Coo and we show that an

operator of class Co is of class Coo • Operators of class Co admit a "local" charac-

terization in the sense that an operator is of class Co if and only if its restriction

to each rationally cyclic invariant subspace (i.e., of the form \J{r(T)x : r G R(Q)}

for some x G H), is of class Co. Finally the last part contains more properties

related to this class of operators, namely a relation between minimal function

and spectrum and relations between minimal functions and rationally invariant

subspaces.

A Jordan operator is a direct sum of Jordan blocks with certain additional

properties, a Jordan block is the compression of the operator of multiplication

by z on

H2(Q)

to the orthogonal complement of a fully invariant subspace. In

the last chapter we achieve a complete characterization of operators of class Co

by showing that each quasisimilarity class contains a unique Jordan operator.

We start by studying operators of class Co having a rationally cyclic vector,

also called multiplicity-free operators. It turns out that each such operator is

quasisimilar to a unique Jordan block, and this is a first step in the classification.

Then we extend the classification results to operators with higher multiplicity.

Many of the ideas and contents of section 4.3 are similar in structure to those

in [7] for the case of the disk. Those proofs which are very similar to the proofs

available in the case of the disk are only briefly sketched.

This paper is based on the author's Indiana University Doctoral Dissertation,

submitted in 1994. The author would like to express her gratitude to her thesis

adviser, Professor Hari Bercovici.