Duality Properties Of Non-Reflexive Bergman Space Of The Upper Half-Plane

Abstract

The study of duality properties of the spaces of analytic functions contin-

ues to attract the attention of many mathematicians. Most studies have

concentrated on the re

exive Hardy and Bergman spaces both on the unit

disk and the upper half-plane. For instance, Zhu, Peloso, among others

have determined the duality properties of Hardy and Bergman spaces.

For the non-re

exive Bergman spaces of the disk, it was proved by Axler

that the dual and the predual are identied as big and little Bloch spaces

respectively. For non-re

exive Bergman spaces of the upper half-plane

L1

a(U; ), the dual is well known as the Bloch space B1(U; i) but the

predual is not known. In our study therefore, we have determined the

predual of L1

a(U; ). We have also determined the group of weighted

composition operators dened on predual space of L1

a(U; ) and inves-

tigated both its semigroup and spectral properties. To determine the

predual space of L1

a(U; ), we used the Cayley transform as well as re-

lated works on the unit disk by Zhu, Peloso among others. To investigate

the properties of the weighted composition groups, we employed func-

tional analysis techniques as well as semigroup theory of linear operators

to determine the innitesimal generator of the semigroup and established

the strong continuity property. Using spectral theory, we determined the

resolvents of the innitesimal generator which were obtained as integral

operators. Finally, we used known theorems like the Hill-Yosida theorem

and spectral mapping theorem to obtain the spectral properties of the

obtained integral operators. The results obtained in this study is of great

importance to the physicists where the concept of semigroup properties

plays a major role in the evolution equations.