  In the context of a theorem of Richter, we establish a similarity between
$C_0$-semigroups of analytic 2-isometries $\{T(t)\}_{t\geq0}$ acting on a
Hilbert space $\mathcal H$ and the multiplication operator semigroup
$\{M_{\phi_t}\}_{t\geq 0}$ induced by $\phi_t(s)=\exp (-st)$ for $s$ in the
right-half plane $\mathbb{C}_+$ acting boundedly on weighted Dirichlet spaces
on $\mathbb{C}_+$. As a consequence, we derive a connection with the right
shift semigroup $\{S_t\}_{t\geq 0}$ $$ S_tf(x)=\left \{ \begin{array}{ll} 0 &
\mbox { if }0\leq t\leq x, \\ f(x-t)& \mbox { if } x>t, \end{array} \right . $$
acting on a weighted Lebesgue space on the half line $\mathbb{R}_+$ and address
some applications regarding the study of the invariant subspaces of
$C_0$-semigroups of analytic 2-isometries.
