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 $L^2({\msbm R}_+)$ 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 invertible, 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 $S_t$.

DOI: 10.14232/actasm-018-546-3

AMS Subject Classification (1991): 47B20, 47B37; 47A10, 46E22

Keyword(s): weighted translation semigroup, completely alternating, completely hyperexpansive, analytic, operator valued weighted shift

Received May 10, 2018 and in final form January 31, 2019. (Registered under 46/2018.)