Semigroups and generators on convex domains with the hyperbolic metric

Reich, Simeon ; Shoikhet, David

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 8 (1997) no. 4, pp. 231-250

Résumé

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 \( B \).