Subnormal weighted translation semigroups

A weighted translation semigroup {S_t} on L²($\text{R}$₊) is defined by $\text{(S}{}_{\mathrm{t}}\text{f)(x) =}\text{(φ(x)}\text{φ(x ? t)}\text{)f(x ? t)}$ for x ? t and 0 otherwise, where φ is a continuous nonzero scalar-valued function on $\text{R}$₊. It is shown that {S_t} is subnormal if and only if φ² is the product of an exponential function and the Laplace-Stieltjes transform of an increasing function of total variation one. A necessary and sufficient condition for similarity of weighted translation semigroups is developed.