C₀-semigroups of 2-isometries and Dirichlet spaces

In the context of a theorem of Richter, we establish a similarity between C₀-semigroups of analytic 2-isometries {T(t)}t≥0 acting on a Hilbert space H and the multiplication operator semigroup {Mϕt}t≥0 induced by ϕt(s)=exp(−st) for s in the right-half plane C₊ acting boundedly on weighted Dirichlet spaces on C₊. As a consequence, we derive a connection with the right shift semigroup {St}t≥0 given by Stf(x)={0 if 0≤x≤t,f(x−t) if x>t, acting on a weighted Lebesgue space on the half line R+ and address some applications regarding the study of the invariant subspaces\linebreak of C₀-semigroups of analytic 2-isometries.