An Analytic Model for left invertible Weighted Translation Semigroups

TL;DR

This paper develops an analytic model for left invertible weighted translation semigroups, representing them as multiplication operators on a vector-valued analytic function space, and explores their spectral properties.

Contribution

It provides a novel analytic model for left invertible weighted translation semigroups as multiplication operators on RKHS, extending the understanding of their structure and spectral characteristics.

Findings

Modeling of left invertible $S_t$ as multiplication by $z$ on RKHS.

Spectral analysis of the weighted translation semigroup.

Application of the model to hyperexpansive semigroups.

Abstract

M. Embry and A. Lambert initiated the study of a semigroup of operators $\{S_t\}$ indexed by a non-negative real number $t$ and termed it as weighted translation semigroup. The operators $S_t$ are defined on $L^2(\mathbb R_+)$ by using a weight function. The operator $S_t$ can be thought of as a continuous analogue of a weighted shift operator. In this paper, we show that every left invertible operator $S_t$ can be modeled as a multiplication by $z$ on a reproducing kernel Hilbert space $\cal H$ of vector-valued analytic functions on a certain disc centered at the origin and the reproducing kernel associated with $\cal H$ is a diagonal operator. As it turns out that every hyperexpansive weighted translation semigroup is left invertile, the model applies to these semigroups. We also describe the spectral picture for the left invertible weighted translation semigroup. In the process, we point out the similarities and differences between a weighted shift operator and an operator $S_t.$