Semigroups and generators on convex domains with the hyperbolic metric (Q1267755)

Summary: Let \(D\) be domain in a complex Banach space \(X\), and let \(\varrho\) 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 \(\varrho\)-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 \(\varrho\) the hyperbolic metric on \(B\). We introduce the notion of a \(\varrho\)-monotone mapping and obtain simple characterizations of generators of semigroups of holomorphic selfmappings of \(B\).