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
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