Rigidity of holomorphic generators and one-parameter semigroups

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dc.creatorElin, M.
dc.creatorLevenshtein, M.
dc.creatorShoikhet, D.
dc.creatorTauraso, R.
dc.date2005-12-21
dc.date.accessioned2026-07-07T06:55:36Z
dc.date.available2026-07-07T06:55:36Z
dc.descriptionIn this paper we establish a rigidity property of holomorphic generators by using their local behavior at a boundary point $τ$ of the open unit disk $Δ$. Namely, if $f\in\mathrm{Hol}(Δ,\mathbb{C})$ is the generator of a one-parameter continuous semigroup $\{F_{t}\}_{t\geq0}$, we state that the equality $f(z)=o(|z-τ|^{3})$ when $z\toτ$ in each non-tangential approach region at $τ$ implies that $f$ vanishes identically on $Δ$. Note, that if $F$ is a self-mapping of $Δ$ then $f=I-F$ is a generator, so our result extends the boundary version of the Schwarz Lemma obtained by D. Burns and S. Krantz. We also prove that two semigroups $\{F_{t}\}_{t\geq0}$ and $\{G_{t}\}_{t\geq0}$, with generators $f$ and $g$ respectively, commute if and only if the equality $f=αg$ holds for some complex constant $α$. This fact gives simple conditions on the generators of two commuting semigroups at their common null point $τ$ under which the semigroups coincide identically on $Δ$.
dc.description20 pages
dc.identifierhttps://arxiv.org/abs/math/0512482
dc.identifierhttp://arxiv.org/abs/math/0512482
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/106329
dc.subjectComplex Variables
dc.subject30D05; 47H20
dc.titleRigidity of holomorphic generators and one-parameter semigroups
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