  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 $S_t.$
