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Angular limits and derivatives for holomorphic maps of infinite dimensional bounded homogeneous domains

Kazimierz Włodarczyk (1994)

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

An infinite dimensional extension of the Pick-Julia theorem is used to derive the conditions of Carathéodory type which guarantee the existence of angular limits and angular derivatives for holomorphic maps of infinite dimensional bounded symmetric homogeneous domains in J * -algebras and in complex Hilbert spaces. The case of operator-valued analytic maps is considered and examples are given.

Approximation de fonctions holomorphes d'un nombre infini de variables

László Lempert (1999)

Annales de l'institut Fourier

Soit X un espace de Banach complexe, et notons B ( R ) X la boule de rayon R centrée en 0 . On considère le problème d’approximation suivant: étant donnés 0 < r < R , ϵ > 0 et une fonction f holomorphe dans B ( R ) , existe-t-il toujours une fonction g , holomorphe dans X , telle que | f - g | < ϵ sur B ( r ) ? On démontre que c’est bien le cas si X est l’espace l 1 des suites sommables.

Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition

Francine Meylan (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let X be a complex Banach space. Recall that X admits afinite-dimensional Schauder decompositionif there exists a sequence { X n } n = 1 of finite-dimensional subspaces of X , such that every x X has a unique representation of the form x = n = 1 x n , with x n X n for every n . The finite-dimensional Schauder decomposition is said to beunconditionalif, for every x X , the series x = n = 1 x n , which represents x , converges unconditionally, that is, n = 1 x π ( n ) converges for every permutation π of the integers. For short, we say that X admits an unconditional F.D.D.We...

Approximation of holomorphic functions of infinitely many variables II

László Lempert (2000)

Annales de l'institut Fourier

Let X be a Banach space and B ( R ) X the ball of radius R centered at 0 . Can any holomorphic function on B ( R ) be approximated by entire functions, uniformly on smaller balls B ( r ) ? We answer this question in the affirmative for a large class of Banach spaces.

Approximation of holomorphic mappings on infinite dimensional spaces.

Erhan Çaliskan (2004)

Revista Matemática Complutense

In this article we examine necessary and sufficient conditions for the predual of the space of holomorphic mappings of bounded type, Gb(U), to have the approximation property and the compact approximation property and we consider when the predual of the space of holomorphic mappings, G(U), has the compact approximation property. We obtain also similar results for the preduals of spaces of m-homogeneous polynomials, Q(mE).

Approximation properties determined by operator ideals and approximability of homogeneous polynomials and holomorphic functions

Sonia Berrios, Geraldo Botelho (2012)

Studia Mathematica

Given an operator ideal ℐ, a Banach space E has the ℐ-approximation property if the identity operator on E can be uniformly approximated on compact subsets of E by operators belonging to ℐ. In this paper the ℐ-approximation property is studied in projective tensor products, spaces of linear functionals, spaces of linear operators/homogeneous polynomials, spaces of holomorphic functions and their preduals.

Averages of holomorphic mappings and holomorphic retractions on convex hyperbolic domains

Simeon Reich, David Shoikhet (1998)

Studia Mathematica

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.

Banach spaces of homogeneous polynomials without the approximation property

Seán Dineen, Jorge Mujica (2015)

Czechoslovak Mathematical Journal

We present simple proofs that spaces of homogeneous polynomials on L p [ 0 , 1 ] and p provide plenty of natural examples of Banach spaces without the approximation property. By giving necessary and sufficient conditions, our results bring to completion, at least for an important collection of Banach spaces, a circle of results begun in 1976 by R. Aron and M. Schottenloher (1976).

Banach-valued axiomatic spectra

S. Seán, Robin E. Harte (2006)

Studia Mathematica

Using axiomatic joint spectra we obtain a functional calculus which extends our previous Gelfand-Waelbroeck type results to include a Banach-valued Taylor-Waelbroeck spectrum.

Bounded analytic sets in Banach spaces

Volker Aurich (1986)

Annales de l'institut Fourier

Conditions are given which enable or disable a complex space X to be mapped biholomorphically onto a bounded closed analytic subset of a Banach space. They involve on the one hand the Radon-Nikodym property and on the other hand the completeness of the Caratheodory metric of X .

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