# Weighted Composition Groups on the Little Bloch space

## TL;DR

This paper analyzes the spectral and semigroup properties of weighted composition groups on the Little Bloch space, revealing their structure as invertible isometries and detailing their generators and resolvents.

## Contribution

It provides a comprehensive analysis of weighted composition groups on the Little Bloch space, including their spectral properties and their relation to automorphisms of the upper half plane.

## Key findings

- Weighted composition groups are strongly continuous invertible isometries.
- The spectra and generators of these groups are explicitly characterized.
- The analysis extends to the adjoint groups on the nonreflexive Bergman space.

## Abstract

We determine both the semigroup and spectral properties of a group of weighted composition operators on the Little Bloch space. It turns out that these are strongly continuous groups of invertible isometries on the Bloch space. We then obtain the norm and spectra of the infinitesimal generator as well as the resulting resolvents which are given as integral operators. As consequences, we complete the analysis of the adjoint composition group on the predual on the nonreflexive Bergman space, and a group of isometries associated with a specific automorphism of the upper half plane.