Reich, Simeon; Shoikhet, David Semigroups and generators on convex domains with the hyperbolic metric. (English) Zbl 0905.47056 Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 8, No. 4, 231-250 (1997). 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 \(B\). Cited in 30 Documents MSC: 47H20 Semigroups of nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47H05 Monotone operators and generalizations Keywords:monotone operator; pseudometric; Schwarz-Pick system; generator of a \(\varrho\)-nonexpansive semigroup; nonlinear resolvent; hyperbolic metric; semigroups of holomorphic selfmappings PDFBibTeX XMLCite \textit{S. Reich} and \textit{D. Shoikhet}, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 8, No. 4, 231--250 (1997; Zbl 0905.47056) Full Text: EuDML