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 identi ed 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 de ned 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 in nitesimal generator of the semigroup and established the strong continuity property. Using spectral theory, we determined the resolvents of the in nitesimal 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.