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.

Soit $X$ un espace de Banach complexe, et notons $B\left(R\right)\subset X$ 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 $B\left(R\right)$, existe-t-il toujours une fonction $g$, holomorphe dans $X$, telle que $|f-g|\<ϵ$ sur $B\left(r\right)$? On démontre que c’est bien le cas si $X$ est l’espace $l^1$ des suites sommables.

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

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

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).

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.

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.