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An Agler-type model theorem for C0-semigroups of Hilbert space contractions

Research paper by Rydhe, E.

Indexed on: 10 May '16Published on: 18 Mar '16Published in: Journal of the London Mathematical Society



Abstract

We investigate suitable conditions for a $C_0$-semigroup $(T(t))_{t\ge 0}$ of Hilbert space contractions to be unitarily equivalent to the restriction of the adjoint shift semigroup $(S_\gamma ^*(t))_{t\ge 0}$ to an invariant subspace of the standard weighted Bergman space $A_{\gamma -2}(\mathbb {C}_+,\mathcal {K})$. It turns out that $(T(t))_{t\ge 0}$ admits a model by $(S_\gamma ^*(t))_{t\ge 0}$ if and only if its cogenerator is $\gamma $-hypercontractive and $\lim _{t\to 0}T(t)=0$ in strong operator topology. We then discuss how such semigroups can be characterized without involving the cogenerator. A sufficient condition is that, for each $t> 0,$ the operator $T(t)$ is $\gamma $-hypercontractive. Surprisingly, this condition is necessary if and only if $\gamma $ is integer. The paper is concluded with a conjecture that would imply a more symmetric characterization.