Generalized Bloch Spaces of the Upper Half-Plane and Their Composition Semigroups

Abstract

Banach space structure of the Bloch space of the unit disc B(D) has been studied widely by many Mathematicians. Cima, Anderson, among others have proved that the Bloch space of the unit disc, B(D) and the little Bloch space of the unit disc, B0(D) are Banach spaces with respect to the Bloch norm. Boundedness, compactness, as well as semigroup properties have been studied on the Bloch spaces of the unit disc. Zhu among other scholars have studied the generalized little Bloch space of the unit disc B_ (D), as closed, separable subspace of the generalized Bloch space of the unit disc B_(D). On the other hand, there is little and much less complete literature on Bloch space of other domains. On the upper half plane, U, the properties of the generalized Bloch spaces as Banach spaces are not known. In our study therefore, we have investigated the properties of the generalized Bloch space of the upper half plane, B_(U). Speci_cally, we have proved that B_(U) and the generalized little Bloch space of the upper half plane, B_ (U) are Banach spaces. Cayley transform has been employed in getting equivalent representation of functions from B_(D) to B_(U). By use of classi_cation theorem for the automorphisms of U, we have established that automorphism groups of U generate strongly continuous semigroups on B__(U). We applied the theory of linear opera-tors in the study of semigroup properties of the composition semigroups. The results of this study have contributed to the existing knowledge and enhanced further research in this _eld of study