Geodesic
Semigroups and generators on convex domains with the hyperbolic metric
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 8 (1997) no. 4, pp. 231-250.

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Let \( D \) be domain in a complex Banach space \( X \), and let \( \rho \) 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 \( f : D \rightarrow X \) to be a generator of a \( \rho \)-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 \( \rho \) the hyperbolic metric on \( B \). We introduce the notion of a \( \rho \)-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 \( \rho \) una pseudometrica assegnata a \( D \) da un sistema di Schwarz-Pick. Nella prima parte del lavoro si stabiliscono alcuni criteri affinché una applicazione \( f : D \rightarrow X \) sia un generatore di un semigruppo \( \rho \)-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 \( \rho \) la metrica iperbolica su \( B \). Si introduce la nozione di applicazione \( \rho \)-monotona e si ottengono semplici caratterizzazioni di generatori di semigruppi di applicazioni olomorfe di \( B \) in sé. 
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Reich, Simeon; Shoikhet, David. Semigroups and generators on convex domains with the hyperbolic metric. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 8 (1997) no. 4, pp. 231-250. http://geodesic.mathdoc.fr/item/RLIN_1997_9_8_4_a0/

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