dc.description.abstract | Semigroups of composition operators on spaces of analytic functions have
interested mathematicians for many years. The breakthrough in Hardy
spaces was given by Berkson and Porta. On Bergman and Dirichlet
spaces, Siskakis did extensive research whereby he determined semigroup
as well as spectral properties of these operators on the unit disk. Later,
other researchers like Arvanitidis, Bonyo, Blasco, Matache and others
extended the work to Hardy and Bergman spaces of the unit disk and
upper half plane. However, very little is known about sernigroups of
weighted composition operators on Bloch spaces. In this study therefore
we investigated the properties of semigroups of weighted composition operators 011 the Bloch space. In particular, we determined a semigroup
of weighted composition operators on the Bloch space; investigated its
semigroup properties; and determined its spectral properties on the Bloch
space of the unit disk. We used the duality properties of the non reflexive Bergman space to identify a semigroup of composition operators. To
obtain the semigroup properties, we employed the theory of semigroups
of linear operators and functional analysis where we determined infinitesimal generator of the semigroup and established the strong continuity
property. We then determined the resolvents of the infinitesimal generator which are obtained as integral operators. Using the spectral mapping
theorems as well as the Hille Yosida theorem, we obtained the spectral
properties of the resulting integral operators. The results of this study
will indeed contribute new knowledge and we hope it will motivate further
research in this area of study. | en_US |