$C_0$-semigroups of 2-isometries and Dirichlet spaces

In the context of a theorem of Richter, we establish a similarity between $C_0$-semigroups of analytic 2-isometries $\{T(t){\}}_{t\ge 0}$ acting on a Hilbert space $\mathcal{H}$ and the multiplication operator semigroup $\{M_{{\varphi }_t}{\}}_{t\ge 0}$ induced by ${\varphi }_t(s)=\exp (-st)$ 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 $\{S_t{\}}_{t\ge 0}$ $$S_{t}f(x)=\{\begin{array}{ll}0& \text{mbox}if0\le t\le x,\\ f(x-t)& \text{mbox}ifx>t,\end{array}$$ 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.