2 ADELE ZUCCHI

case of the disk.

The influence of Sz.-Nagy Dilation Theorem on the development of operator

theory has been extraordinary. Indeed, it removed much of the mistery that

had surrounded von Neumann's Inequality by establishing a simple geometric

explanation. Moreover it prompted a number of mathematicians to ask the

following question:

Let F C C be a compact set and let T G C{H). Are the following two conditions

equivalent?

(i) F is a spectral set for T.

(ii) There exists a Hilbert space K 5 H and a normal operator N G C(K)

such that J(N) C dF and if PH denotes the orthogonal projection of K onto JJT,

then f(T) = PHf(N)\H, for all / G Rat(F).

In [12] R.C. Douglas and V. Paulsen show that if F is sufficiently "nice"

(roughly, Rat(F) is hypo-Dirichlet) and S G C{H) has F as spectral set, then S

is similar to an operator T G C(H) satisfying (ii). In [2] J. Agler proves that the

answer to this question is yes if F is an annulus, but for general compact sets F

the question remains unsolved to this day. For this reason we can not use the

powerful tool of the existence of a normal dilation.

An important tool for our work, and in general for the study of Hilbert space

operators related to multiply connected regions, is the characterization of fully

invariant subspaces of

HP(Q).

This characterization closely resembles Beurling's

characterization of the shift invariant subspaces for the Hardy spaces on the

disk. Sarason [22] provided the description in the case where O is an annulus,

and Hasumi [16] and Voichick [30, 31] for more general regions.

What makes Co operators so well-understood is that their properties are

closely related with the arithmetic of H°°(Q). Our work is therefore partially

devoted to the study of functions in iJ°°(0). There are two approaches to func-

tion theory on multiply connected regions. The first is to work directly on the

region and examine the analytic functions in their own customary setting. The

second approach is to "lift" the function theory of the region to the unit disk by

means of a covering projection map. This technique has clear advantages, but

new difficulties arise with the requirement that the functions must be invariant

under the group of linear fractional transformations that fix the covering map.

We adopt the first approach, which we find more suitable for our purposes.

We use single-valued holomorphic functions. In order to accomplish this, we

shall often need to insert into our formulae harmonic functions continuous up to

the boundary F and constant on each boundary component. This will mean that

inner functions, etc., are only required to have moduli which are constant almost

everywhere on each boundary component of O, rather than having those which

are one almost everywhere on I\ Sarason [23], Hasumi [16], Voichick [30, 31] and

others take a different approach, and allow their functions to be multivalued, but

restrict inner functions, etc., to those whose boundary values are 1 in modulus

almost everywhere.

This paper consists of four chapters, the first of which is the introduction.

In the second we recall some of the results about operators of class Co over the

unit disk. We only give a short summary of the results we need, since most of

them will be generalized in chapters 3 and 4. Then we present Hardy spaces on