An analytic model for left invertible weighted translation semigroups

Abstract

M. Embry and A. Lambert initiated the study of a semigroup of operators {St} indexed by a non-negative real number t and termed it as weighted translation semigroup. The operators St are defined on L 2 (R+) by using a weight function. The operator St can be thought of as a continuous analogue of a weighted shift operator. In this paper, we show that every left invertible operator St can be modeled as a multiplication by z on a reproducing kernel Hilbert space H of vector-valued analytic functions on a certain disc centered at the origin and the reproducing kernel associated with 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 St.