$C_0$-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 S_tf(x)=\left \{ \begin{array}{ll} 0 & \text { if }0\leq x\leq t, \\ f(x-t)& \text{ 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.