Semigroups and generators on convex domains with the hyperbolic metric

Simeon Reich; David Shoikhet

Displaying similar documents to “Semigroups and generators on convex domains with the hyperbolic metric”

Metric domains, holomorphic mappings and nonlinear semigroups.

Reich, Simeon, Shoikhet, David (1998)

Abstract and Applied Analysis

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Generation theory for semigroups of holomorphic mappings in Banach spaces.

Reich, Simeon, Shoikhet, David (1996)

Abstract and Applied Analysis

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Averages of holomorphic mappings and holomorphic retractions on convex hyperbolic domains

Simeon Reich, David Shoikhet (1998)

Studia Mathematica

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Let D be a hyperbolic convex domain in a complex Banach space. Let the mapping F ∈ Hol(D,D) be bounded on each subset strictly inside D, and have a nonempty fixed point set ℱ in D. We consider several methods for constructing retractions onto ℱ under local assumptions of ergodic type. Furthermore, we study the asymptotic behavior of the Cesàro averages of one-parameter semigroups generated by holomorphic mappings.

On the Rogosinski radius for holomorphic mappings and some of its applications

Lev Aizenberg, Mark Elin, David Shoikhet (2005)

Studia Mathematica

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The well known theorem of Rogosinski asserts that if the modulus of the sum of a power series is less than 1 in the open unit disk: $| \sum_{n = 0}^{\infty} a ₙ z ⁿ | < 1$ , |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: $| \sum_{n = 0}^{k} a ₙ z ⁿ | < 1$ , |z| < 1/2, and this radius is sharp. We present a generalization of this theorem to holomorphic mappings of the open unit ball into an arbitrary convex domain. Other multidimensional analogs of Rogosinski’s theorem as well as some applications to dynamical systems are...

Fixed points of Lipschitzian semigroups in Banach spaces

Jarosław Górnicki (1997)

Studia Mathematica

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We prove the following theorem: Let p > 1 and let E be a real p-uniformly convex Banach space, and C a nonempty bounded closed convex subset of E. If $T = T_{s} : C \to C : s \in G = [0, \infty)$ is a Lipschitzian semigroup such that $g = l i m i n f_{G ∋ α \to \infty} i n f_{G ∋ δ \geq 0} 1 / α ʃ_{0}^{α} ∥ T_{β + δ} ∥^{p} d β < 1 + c$ , where c > 0 is some constant, then there exists x ∈ C such that $T_{s} x = x$ for all s ∈ G.

Infinite products of holomorphic mappings.

Budzyńska, Monika, Reich, Simeon (2005)

Abstract and Applied Analysis

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Extensions of nonexpansive mappings in the Hilbert ball with the hyperbolic metric. I.

Tadeusz Kuczumow, Adam Stachura (1988)

Commentationes Mathematicae Universitatis Carolinae

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Converging semigroups of holomorphic maps

Marco Abate (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper we study the semigroups $\Phi:\mathbb{R}^{+}\rightarrow Hol(D,D)$ of holomorphic maps of a strictly convex domain $D\subset\mathbf{C}^{n}$ into itself. In particular, we characterize the semigroups converging, uniformly on compact subsets, to a holomorphic map $h:D\rightarrow\mathbf{C}^{n}$ .

A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces

Viêt-Anh Nguyên (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$ ) be a non locally pluripolar subset of $X$ (resp. $Y$ ). We show that every separately holomorphic mapping $f : W : = (A \times Y) \cup (X \times B) \to Z$ extends to a holomorphic mapping $\hat{f}$ on $\hat{W} : = \{(z, w) \in X \times Y : \tilde{ω} (z, A, X) + \tilde{ω} (w, B, Y) < 1\}$ such that $\hat{f} = f$ on $W \cap \hat{W},$ where $\tilde{ω} (\cdot, A, X)$ (resp. $\tilde{ω} (\cdot, B, Y))$ is the plurisubharmonic measure of $A$ (resp. $B$ ) relative to $X$ (resp. $Y$ ). Generalizations...