C0C_0-semigroups of 22-isometries and Dirichlet spaces

  • Eva A. Gallardo-Gutiérrez

    Universidad Complutense de Madrid, Spain and Instituto de Ciencias Matemáticas, Madrid, Spain
  • Jonathan R. Partington

    University of Leeds, UK
$C_0$-semigroups of $2$-isometries and Dirichlet spaces cover
Download PDF

A subscription is required to access this article.


In the context of a theorem of Richter, we establish a similarity between C0C_0-semigroups of analytic 22-isometries {T(t)}t0\{T(t)\}_{t\geq0} acting on a Hilbert space H\mathcal H and the multiplication operator semigroup {Mϕt}t0\{M_{\phi_t}\}_{t\geq 0} induced by ϕt(s)=exp(st)\phi_t(s)=\mathrm {exp}(-st) for ss in the right-half plane C+\mathbb{C}_+ acting boundedly on weighted Dirichlet spaces on C+\mathbb{C}_+. As a consequence, we derive a connection with the right shift semigroup {St}t0\{S_t\}_{t\geq 0} given by

Stf(x)={0\mboxif0xt,f(xt)\mboxifx>t,S_tf(x)=\left \{ \begin{array}{ll} 0 & \mbox { if }0\leq x\leq t, \\ f(x-t)& \mbox { if } x>t, \end{array} \right .

acting on a weighted Lebesgue space on the half line R+\mathbb{R}_+ and address some applications regarding the study of the invariant subspaces\linebreak of C0C_0-semigroups of analytic 2-isometries.

Cite this article

Eva A. Gallardo-Gutiérrez, Jonathan R. Partington, C0C_0-semigroups of 22-isometries and Dirichlet spaces. Rev. Mat. Iberoam. 34 (2018), no. 3, pp. 1415–1425

DOI 10.4171/RMI/1030