Semigroups and generators on convex domains with the hyperbolic metric

Reich, Simeon; Shoikhet, David

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 8 (1997), p. 231-250 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

Let $D$ be domain in a complex Banach space $X$ , and let $ρ$ be a pseudometric assigned to $D$ by a Schwarz-Pick system. In the first section of the paper we establish several criteria for a mapping $f : D \to X$ to be a generator of a $ρ$ -nonexpansive semigroup on $D$ in terms of its nonlinear resolvent. In the second section we let $X = H$ be a complex Hilbert space, $D = B$ the open unit ball of $H$ , and $ρ$ the hyperbolic metric on $B$ . We introduce the notion of a $ρ$ -monotone mapping and obtain simple characterizations of generators of semigroups of holomorphic self-mappings of $B$ .

Sia $D$ un dominio in uno spazio di Banach complesso $X$ e sia $ρ$ una pseudometrica assegnata a $D$ da un sistema di Schwarz-Pick. Nella prima parte del lavoro si stabiliscono alcuni criteri affinché una applicazione $f : D \to X$ sia un generatore di un semigruppo $ρ$ -non espansivo su $D$ . Nella seconda parte si suppone che sia $X = H$ , spazio di Hilbert complesso, che $D = B$ disco unitario aperto di $H$ e che sia $ρ$ la metrica iperbolica su $B$ . Si introduce la nozione di applicazione $ρ$ -monotona e si ottengono semplici caratterizzazioni di generatori di semigruppi di applicazioni olomorfe di $B$ in sé.

Publié le : 1997-12-01

MR 1631605 | Zbl 0905.47056

@article{RLIN_1997_9_8_4_231_0,
     author = {Simeon Reich and David Shoikhet},
     title = {Semigroups and generators on convex domains with the hyperbolic metric},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {8},
     year = {1997},
     pages = {231-250},
     zbl = {0905.47056},
     mrnumber = {1631605},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1997_9_8_4_231_0}
}

Reich, Simeon; Shoikhet, David. Semigroups and generators on convex domains with the hyperbolic metric. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 8 (1997) pp. 231-250. http://gdmltest.u-ga.fr/item/RLIN_1997_9_8_4_231_0/

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