The bounded approximation property for the predual of the space of bounded holomorphic mappings

Erhan Çalışkan

Displaying similar documents to “The bounded approximation property for the predual of the space of bounded 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...

Approximation polynômiale dans des classes de jets

Moulay Taïb Belghiti, Boutayeb El Ammari, Laurent P. Gendre (2015)

Banach Center Publications

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In this paper we obtain results on approximation, in the multidimensional complex case, of functions from $^{\infty} (K)$ by complex polynomials. In particular, we generalize the results of Pawłucki and Pleśniak (1986) for the real case and of Siciak (1993) in the case of one complex variable. Furthermore, we extend the results of Baouendi and Goulaouic (1971) who obtained the order of approximation in the case of Gevrey classes over real compacts with smooth analytic boundary and we present the orders...

Holomorphic series expansion of functions of Carleman type

Taib Belghiti (2004)

Annales Polonici Mathematici

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Let f be a holomorphic function of Carleman type in a bounded convex domain D of the plane. We show that f can be expanded in a series f = ∑ₙfₙ, where fₙ is a holomorphic function in Dₙ satisfying $s u p_{z \in D ₙ} | f ₙ (z) | \leq C ϱ ⁿ$ for some constants C > 0 and 0 < ϱ < 1, and where (Dₙ)ₙ is a suitably chosen sequence of decreasing neighborhoods of the closure of D. Conversely, if f admits such an expansion then f is of Carleman type. The decrease of the sequence Dₙ characterizes the smoothness of f. ...

Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition

Francine Meylan (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $X$ be a complex Banach space. Recall that $X$ admits aif there exists a sequence ${X_{n}}_{n = 1}^{\infty}$ 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 beif, for every $x \in X,$ the series $x = \sum_{n = 1}^{\infty} x_{n},$ which represents $x,$ converges unconditionally, that is, $\sum_{n = 1}^{\infty} x_{π (n)}$ converges for every permutation $π$ of the integers. For short, we say that $X$ admits an unconditional F.D.D.We show that if X admits an unconditional...

Banach spaces of homogeneous polynomials without the approximation property

Seán Dineen, Jorge Mujica (2015)

Czechoslovak Mathematical Journal

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We present simple proofs that spaces of homogeneous polynomials on $L_{p} [0, 1]$ and $ℓ_{p}$ 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).

A unified approach to the theory of separately holomorphic mappings

Viêt-Anh Nguyên (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We extend the theory of separately holomorphic mappings between complex analytic spaces. Our method is based on Poletsky theory of discs, Rosay theorem on holomorphic discs and our recent joint-work with Pflug on boundary cross theorems in dimension $1 .$ It also relies on our new technique of conformal mappings and a generalization of Siciak’s relative extremal function. Our approach illustrates the unified character: “From local information to global extensions”. Moreover, it avoids systematically...

A set on which the local Łojasiewicz exponent is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

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Let U be a neighbourhood of 0 ∈ ℂⁿ. We show that for a holomorphic mapping $F = (f ₁, . . ., f ₘ) : U \to ℂ^{m}$ , F(0) = 0, the Łojasiewicz exponent ₀(F) is attained on the set z ∈ U: f₁(z)·...·fₘ(z) = 0.

Schauder bases and the bounded approximation property in separable Banach spaces

Jorge Mujica, Daniela M. Vieira (2010)

Studia Mathematica

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Let E be a separable Banach space with the λ-bounded approximation property. We show that for each ϵ > 0 there is a Banach space F with a Schauder basis such that E is isometrically isomorphic to a 1-complemented subspace of F and, moreover, the sequence (Tₙ) of canonical projections in F has the properties $s u p_{n \in ℕ} | | T ₙ | | \leq λ + ϵ$ and $l i m s u p_{n \to \infty} | | T ₙ | | \leq λ$ . This is a sharp quantitative version of a classical result obtained independently by Pełczyński and by Johnson, Rosenthal and Zippin.

Certain partial differential subordinations on some Reinhardt domains in $ℂ^{n}$

Gabriela Kohr, Mirela Kohr (1997)

Annales Polonici Mathematici

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We obtain an extension of Jack-Miller-Mocanu’s Lemma for holomorphic mappings defined in some Reinhardt domains in $ℂ^{n}$ . Using this result we consider first and second order partial differential subordinations for holomorphic mappings defined on the Reinhardt domain $B_{2 p}$ with p ≥ 1.

An extension theorem for separately holomorphic functions with analytic singularities

Marek Jarnicki, Peter Pflug (2003)

Annales Polonici Mathematici

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Let $D_{j} \subset ℂ^{k_{j}}$ be a pseudoconvex domain and let $A_{j} \subset D_{j}$ be a locally pluriregular set, j = 1,...,N. Put $X : = ⋃_{j = 1}^{N} A ₁ \times . . . \times A_{j - 1} \times D_{j} \times A_{j + 1} \times . . . \times A_{N} \subset ℂ^{k ₁ + . . . + k_{N}}$ . Let U be an open connected neighborhood of X and let M ⊊ U be an analytic subset. Then there exists an analytic subset M̂ of the “envelope of holomorphy” X̂ of X with M̂ ∩ X ⊂ M such that for every function f separately holomorphic on X∖M there exists an f̂ holomorphic on X̂∖M̂ with ${f ̂ |}_{X ∖ M} = f$ . The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001]. ...

Contracting rigid germs in higher dimensions

Matteo Ruggiero (2013)

Annales de l’institut Fourier

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Following Favre, we define a holomorphic germ $f : (ℂ^{d}, 0) \to (ℂ^{d}, 0)$ to be rigid if the union of the critical set of all iterates has simple normal crossing singularities. We give a partial classification of contracting rigid germs in arbitrary dimensions up to holomorphic conjugacy. Interestingly enough, we find new resonance phenomena involving the differential of $f$ and its linear action on the fundamental group of the complement of the critical set.

On a class of inner maps

Edoardo Vesentini (2005)

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

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Let $f$ be a continuous map of the closure $\bar{Δ}$ of the open unit disc $Δ$ of $C$ into a unital associative Banach algebra $A$ , whose restriction to $Δ$ is holomorphic, and which satisfies the condition whereby $0 \notin σ (f (z)) \subset \bar{Δ}$ for all $z \in Δ$ and $σ (f (z)) \subset \partial Δ$ whenever $z \in \partial Δ$ (where $σ (x)$ is the spectrum of any $x \in A$ ). One of the basic results of the present paper is that $f$ is , that is to say, $σ (f (z))$ is then a compact subset of $\partial Δ$ that does not depend on $z$ for all $z \in \bar{Δ}$ . This fact will be applied to holomorphic self-maps of the open unit ball of some...

The non-pluripolarity of compact sets in complex spaces and the property $(L B^{\infty})$ for the space of germs of holomorphic functions

Le Mau Hai, Tang Van Long (2002)

Studia Mathematica

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The aim of this paper is to establish the equivalence between the non-pluripolarity of a compact set in a complex space and the property $(L B^{\infty})$ for the dual space of the space of germs of holomorphic functions on that compact set.

Proper holomorphic self-mappings of the minimal ball

Nabil Ourimi (2002)

Annales Polonici Mathematici

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The purpose of this paper is to prove that proper holomorphic self-mappings of the minimal ball are biholomorphic. The proof uses the scaling technique applied at a singular point and relies on the fact that a proper holomorphic mapping f: D → Ω with branch locus $V_{f}$ is factored by automorphisms if and only if $f_{*} (π ₁ (D ∖ f^{- 1} (f (V_{f})), x))$ is a normal subgroup of $π ₁ (Ω ∖ f (V_{f}), b)$ for some $b \in Ω ∖ f (V_{f})$ and $x \in f^{- 1} (b)$ .

Analytic cohomology of complete intersections in a Banach space

Imre Patyi (2004)

Annales de l’institut Fourier

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Let $X$ be a Banach space with a countable unconditional basis (e.g., $X = ℓ_{2}$ ), $Ω \subset X$ an open set and $f_{1}, ..., f_{k}$ complex-valued holomorphic functions on $Ω$ , such that the Fréchet differentials $d f_{1} (x), ..., d f_{k} (x)$ are linearly independant over $ℂ$ at each $x \in Ω$ . We suppose that $M = {x \in Ω : f_{1} (x) = ... = f_{k} (x) = 0}$ is a complete intersection and we consider a holomorphic Banach vector bundle $E \to M$ . If $I$ (resp. $𝒪^{E}$ ) denote the ideal of germs of holomorphic functions on $Ω$ that vanish on $M$ (resp. the sheaf of germs of holomorphic sections of $E$ ), then the sheaf cohomology groups...