$C_0$-semigroups of 2-isometries and Dirichlet spaces (1806.06816v1)

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 $\phi_t(s)=\exp (-st)$ 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.