Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 33, 2008, 231-240

ALEKSANDROV-CLARK MEASURES AND SEMIGROUPS OF ANALYTIC FUNCTIONS IN THE UNIT DISC

Filippo Bracci, Manuel D. Contreras and Santiago Díaz-Madrigal

Università di Roma "Tor Vergata", Dipartimento Di Matematica
Via Della Ricerca Scientifica 1, 00133, Roma, Italy; fbracci 'at' mat.uniroma2.it

Universidad de Sevilla, Escuela Técnica Superior de Ingenieros, Departamento de Matemática Aplicada II
Camino de los Descubrimientos, s/n, 41092, Sevilla, Spain; contreras 'at' us.es

Universidad de Sevilla, Escuela Técnica Superior de Ingenieros, Departamento de Matemática Aplicada II
Camino de los Descubrimientos, s/n, 41092, Sevilla, Spain; madrigal 'at' us.es

Abstract. In this paper we prove a formula describing the infinitesimal generator of a continuous semigroup (\varphi_t) of holomorphic self-maps of the unit disc with respect to a boundary regular fixed point. The result is based on Aleksandrov-Clark measures techniques. In particular we prove that the Aleksandrov-Clark measure of (\varphi_t) at a boundary regular fixed point is differentiable (in the weak^*-topology) with respect to t.

2000 Mathematics Subject Classification: Primary 30E20, 30D40.

Key words: Angular derivative, Aleksandrov-Clark measure, semigroups of analytic functions.

Reference to this article: F. Bracci, M.D. Contreras and S. Díaz-Madrigal: Aleksandrov-Clark measures and semigroups of analytic functions in the unit disc. Ann. Acad. Sci. Fenn. Math. 33 (2008), 231-240.

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