Revista Matemática Iberoamericana

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Volume 34, Issue 3, 2018, pp. 1415–1425
DOI: 10.4171/RMI/1030

Published online: 2018-08-27

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

Eva A. Gallardo-Gutiérrez[1] and Jonathan R. Partington[2]

(1) Universidad Complutense de Madrid, Spain and Instituto de Ciencias Matemáticas, Madrid, Spain
(2) University of Leeds, UK

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 $$S_tf(x)=\left \{ \begin{array}{ll} 0 & \mbox { if }0\leq x\leq t, \\ f(x-t)& \mbox { if } x>t, \end{array} \right .$$ acting on a weighted Lebesgue space on the half line $\mathbb{R}_+$ and address some applications regarding the study of the invariant subspaces\linebreak of $C_0$-semigroups of analytic 2-isometries.

Keywords: 2-isometries, right-shift semigroups, Dirichlet space

Gallardo-Gutiérrez Eva, Partington Jonathan: $C_0$-semigroups of $2$-isometries and Dirichlet spaces. Rev. Mat. Iberoamericana 34 (2018), 1415-1425. doi: 10.4171/RMI/1030