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

TL;DR

This paper establishes a similarity between $C_0$-semigroups of analytic 2-isometries and multiplication operator semigroups on weighted Dirichlet spaces, linking them to right shift semigroups and exploring invariant subspaces.

Contribution

It proves a similarity theorem connecting $C_0$-semigroups of 2-isometries with multiplication operators on Dirichlet spaces, extending Richter’s theorem.

Findings

Established similarity between 2-isometry semigroups and multiplication operators.

Connected these semigroups to right shift semigroups on weighted Lebesgue spaces.

Addressed applications to invariant subspaces of these semigroups.

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.