| dc.description.abstract | Composition groups has been a topic of interest in the past decades.
Most studies have been on spaces of analytic functions of the unit disk.
For instance; Matache studied bouncledness and compactness of composition
operators; Berkson and Porta delve on the structure of semigroups
of functions and their basic properties, on Hardy and Bergman spaces
of the disk. For the Dirichlet space of the unit disk, Siskakis proved
strong continuity of sernigroups and compactness of the resolvent operator.
Bonyo undertook spectral analysis of certain groups of isometries on
Hardy and Bergman spaces of the upper half plane. Little has been done
OIl the Dirichlet space of the upper-half plane and this formed the basis
of our study. In this thesis, we determined the composition groups illduced
by the scaling, translation and rotation groups; investigated. both
the semigroup as well as the spectral properties of each group on the
Dirichlet space of the upper-half plane. To determine the composition
groups, known definitions of weighted composition operators as well as the
semigroup theory of linear operators on Banach spaces were used. To investigate
the semigroup properties, the infinitesimal generators and their
domains, the strong continuity property for each group were determined.
For the rotation group, we applied the theory of similar semigroups to
carry out a complete spectral analysis ofthe composition group as well as
the resulting resolvents which were obtained as integral operators. The
results of this study add reasonably to the existing literature and are useful
in advancement of research in this area and ill optimal control theory
where integral equations and integral operators are usually applied. | en_US |
