Duality of the nonreflexive Bergman space of the upper half plane and Composition groups

TL;DR

This paper characterizes the predual of a nonreflexive Bergman space on the upper half plane as a little Bloch space and studies the semigroup and spectral properties of associated composition operator groups.

Contribution

It identifies the predual of the nonreflexive Bergman space with the little Bloch space and analyzes the properties of the resulting composition operator groups.

Findings

Predual of the nonreflexive Bergman space is the little Bloch space vanishing at i.

Analyzes semigroup properties of composition operator adjoints.

Examines spectral properties of the composition groups.

Abstract

We identify the predual of the nonreflexive Bergman space of the upper half plane, $L_a^1(\uP,\mu_{\al})$, with the little Bloch space of the upper half plane consisting of functions vanishing at $i$. We then investigate both the semigroup and spectral properties of the adjoint groups of composition operators which are naturally obtained from the duality pairing and are therefore defined on the identified predual.