Reproducing Kernels for Hardy and Bergman Spaces of the Upper Half Plane

Abstract

Using invertible isometries between Hardy and Bergman spaces of the unit disk D and the corresponding spaces of the upper half plane U, we determine explicitly the reproducing kernels for the Hardy and Bergman spaces of U. As a consequence, we obtain the duality relations for the reflexive Hardy and Bergman spaces of the half plane U.

Composition-Differentiation Operator on the Bergman Space

“…See [1,2,11] and references therein for more details. Let ψ be a self analytic map on D, then the composition operator C ψ induced by ψ and acting on H(D) is defined by:…”