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

Abstract

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 $\phi_t(s)=\mathrm {exp}(-st)$ 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 $\{S_t\}_{t\geq 0}$ given by

acting on a weighted Lebesgue space on the half line $\mathbb{R}_+$ and address some applications regarding the study of the invariant subspaces\linebreak of $C_0$-semigroups of analytic 2-isometries.