Dirichlet approximation and universal Dirichlet series

We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical results of Runge, Mergelyan and Vitushkin. We also strengthen the notion of universal Dirichlet series.

Weakly continuous mappings on Banach spaces

Abstract It is shown that every n -homogeneous continuous polynomial on a Banach space E which is weakly continuous on the unit ball of E is weakly uniformly continuous on the unit ball of E . Applications of the result to spaces of polynomials and holomorphic mappings on E are given.

Extendability and domains of holomorphy in infinite-dimensional spaces

On norm attaining polynomials

We show that for every Banach space X the set of 2-homogeneous continuous polynomials whose canonical extension to X∗∗ attain their norm is a dense subset of the space of all 2-homogeneous continuous polynomials P(2X).

Weakly compact multilinear mappings

The notion of Arens regularity of a bilinear form on a Banach space E is extended to continuous m-linear forms, in such a way that the natural associated linear mappings, E→L (m−1E) and (m – l)-linear mappings E × … × E → E', are all weakly compact. Among other applications, polynomials whose first derivative is weakly compact are characterized.

Cluster values of holomorphic functions of bounded type

We study the cluster value theorem for Hb(X), the Fréchet algebra of holomorphic functions bounded on bounded sets of X. We also describe the (size of) fibers of the spectrum of Hb(X). Our results are rather complete whenever X has an unconditional shrinking basis and for X = ℓ1. As a byproduct, we obtain results on the spectrum of the algebra of all uniformly continuous holomorphic functions on the ball of ℓ1. Fil: Aron, Richard Martin. Kent State University; Estados Unidos Fil: Carando, Daniel Germán. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas ; Argentina Fil: Lassalle, S…

The Bishop–Phelps–Bollobás theorem for operators

AbstractWe prove the Bishop–Phelps–Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop–Phelps–Bollobás theorem holds for operators from ℓ1 into Y. Several examples of classes of such spaces are provided. For instance, the Bishop–Phelps–Bollobás theorem holds when the range space is finite-dimensional, an L1(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.

The Bishop–Phelps–Bollobás theorem for L(L1(μ),L∞[0,1])

Abstract We show that the Bishop–Phelps–Bollobas theorem holds for all bounded operators from L 1 ( μ ) into L ∞ [ 0 , 1 ] , where μ is a σ-finite measure.

Gleason parts for algebras of holomorphic functions on the ball of $\mathbf{c_0}$

For a complex Banach space $X$ with open unit ball $B_X,$ consider the Banach algebras $\mathcal H^\infty(B_X)$ of bounded scalar-valued holomorphic functions and the subalgebra $\mathcal A_u(B_X)$ of uniformly continuous functions on $B_X.$ Denoting either algebra by $\mathcal A,$ we study the Gleason parts of the set of scalar-valued homomorphisms $\mathcal M(\mathcal A)$ on $\mathcal A.$ Following remarks on the general situation, we focus on the case $X = c_0.$

Connected components in the space of composition operators onH∞ functions of many variables

LetE be a complex Banach space with open unit ballBe. The structure of the space of composition operators on the Banach algebra H∞, of bounded analytic functions onBe with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly insideBe form a path connected component. WhenE is a Hilbert space or aCo(X)- space, the path connected components are shown to be the open balls of radius 2.

Reflexivity and nonweakly null maximizing sequences

We introduce and explore a new property related to reflexivity that plays an important role in the characterization of norm attaining operators. We also present an application to the theory of compact perturbations of linear operators and characterize norm attaining scalar-valued continuous 2 2 -homogeneous polynomials on ℓ 2 \ell _{2} .

Homomorphisms on spaces of weakly continuous holomorphic functions

Let X be a Banach space and let $X^{\ast }$ be its topological dual space. We study the algebra ${\cal H}_{w^\ast}(X^{\ast})$ of entire functions on $X^{\ast }$ that are weak-star continuous on bounded sets. We prove that every m-homogeneous polynomial of finite type P on $X^*$ that is weak-star continuous on bounded sets can be written in the form $P=\textstyle\sum\limits _{j=1}^q x_{1j}\cdots x_{mj}$ where $x_{ij} \in X$ , for all i,j. As an application, we characterize convolution homomorphisms on ${\cal H}_{w^\ast}(X^{\ast})$ and on the space ${\cal H}_{wu}(X)$ of entire functions on X which are weakly uniformly continuous on bounded subsets of X, assuming that X * has the approximation…

CHAOTIC POLYNOMIALS IN SPACES OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS

AbstractWe construct chaotic m-homogeneous maps acting on $\mathcal{C}^{r}_{\mathtt{+}}( [0,\infty ))$ for any m ≥ 2, $r\in\mathbb{N}\cup\{0\},$ and on the Fréchet spaces $\mathcal{C}_{\mathbb{R}}(\mathbb{R})$ for odd values of m ≥ 3 and $\mathcal{C}_{\mathbb{C}}(\mathbb{R})$ for any m ≥ 2.

ALGEBRAS OF SYMMETRIC HOLOMORPHIC FUNCTIONS ON ${\cal L}_p$

Group-symmetric holomorphic functions on a Banach space

We study the holomorphic functions on a complex Banach space E that are invariant under the action of a given group of operators on E. A great variety of situations occur depending, of course, on the group and the space. Nevertheless, in the examples we deal with, they can be described in terms of a few natural ones and functions of a finite number of variables. Fil: Aron, Richard. Universidad de Valencia; España Fil: Galindo, Pablo. Universidad de Valencia; España Fil: Pinasco, Damian. Universidad Torcuato Di Tella; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Zalduendo, Ignacio Martin. Universidad Torcuato Di Tella; Argentina. Consejo Nacional de I…

A multilinear Phelps' Lemma

We prove a multilinear version of Phelps' Lemma: if the zero sets of multilinear forms of norm one are 'close', then so are the multilinear forms.

Smooth surjections and surjective restrictions

Given a surjective mapping $f : E \to F$ between Banach spaces, we investigate the existence of a subspace $G$ of $E$, with the same density character as $F$, such that the restriction of $f$ to $G$ remains surjective. We obtain a positive answer whenever $f$ is continuous and uniformly open. In the smooth case, we deduce a positive answer when $f$ is a $C^1$-smooth surjection whose set of critical values is countable. Finally we show that, when $f$ takes values in the Euclidean space $\mathbb R^n$, in order to obtain this result it is not sufficient to assume that the set of critical values of $f$ has zero-measure.

The Bishop–Phelps–Bollobás theorem for L(L1(μ),L∞[0,1])

AbstractWe show that the Bishop–Phelps–Bollobás theorem holds for all bounded operators from L1(μ) into L∞[0,1], where μ is a σ-finite measure.

Regularity and Algebras of Analytic Functions in Infinite Dimensions

A Banach space E E is known to be Arens regular if every continuous linear mapping from E E to E ′ E’ is weakly compact. Let U U be an open subset of E E , and let H b ( U ) H_b(U) denote the algebra of analytic functions on U U which are bounded on bounded subsets of U U lying at a positive distance from the boundary of U . U. We endow H b ( U ) H_b(U) with the usual Fréchet topology. M b ( U ) M_b(U) denotes the set of continuous homomorphisms ϕ : H b ( U ) → C \phi :H_b(U) \to \mathbb {C} . We study the relation between the Arens regularity of the space E E and the structure of M b ( U ) M_b(U) .

Behavior of holomorphic mappings on $p$-compact sets in a Banach space

We study the behavior of holomorphic mappings on p-compact sets in Banach spaces. We show that the image of a p-compact set by an entire mapping is a p-compact set. Some results related to the localization of p-compact sets in the predual of homogeneous polynomials are also obtained. Finally, the "size" of p-compactness of the image of the unit ball by p-compact linear operators is studied.