Norm, essential norm and weak compactness of weighted composition operators between dual Banach spaces of analytic functions

Abstract In this paper we estimate the norm and the essential norm of weighted composition operators from a large class of – non-necessarily reflexive – Banach spaces of analytic functions on the open unit disk into weighted type Banach spaces of analytic functions and Bloch type spaces. We also show the equivalence of compactness and weak compactness of weighted composition operators from these weighted type spaces into a class of Banach spaces of analytic functions, that includes a large family of conformally invariant spaces like BMOA and analytic Besov spaces.

Entire Functions of Bounded Type on Fréchet Spaces

We show that holomorphic mappings of bounded type defined on Frechet spaces extend to the bidual. The relationship between holomorphic mappings of bounded type and of uniformly bounded type is discussed and some algebraic and topological properties of the space of all entire mappings of (uniformly) bounded type are proved, for example a holomorphic version of Schauder's theorem.

Fredholm composition operators on algebras of analytic functions on Banach spaces

AbstractWe prove that Fredholm composition operators acting on the uniform algebra H∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.

Königs eigenfunction for composition operators on Bloch and H∞ type spaces

Abstract We discuss when the Konigs eigenfunction associated with a non-automorphic selfmap of the complex unit disc that fixes the origin belongs to Banach spaces of holomorphic functions of Bloch and H ∞ type. In the latter case, our characterization answers a question of P. Bourdon. Some spectral properties of composition operators on H ∞ for unbounded Konigs eigenfunction are obtained.

A stronger Dunford-Pettis property

Weakly compact composition operators between algebras of bounded analytic functions

Holomorphic mappings of bounded type

Abstract For a Banach space E, we prove that the Frechet space H b(E) is the strong dual of an (LB)-space, B b(E), which leads to a linearization of the holomorphic mappings of bounded type. It is also shown that the holomorphic functions defined on (DFC)-spaces are of uniformly bounded type.

Compact and Weakly Compact Homomorphisms on Fréchet Algebras of Holomorphic Functions

We study homomorphisms between Frechet algebras of holomorphic functions of bounded type. In this setting we prove that any pointwise bounded homomorphism into the space of entire functions of bounded type is rank one. We characterize up to the approximation property of the underlying Banach space, the weakly compact composition operators on Hb(V), V absolutely convex open set.

Grothendieck-type subsets of Banach lattices

Abstract In the setting of Banach lattices the weak (resp. positive) Grothendieck spaces have been defined. We localize such notions by defining new classes of sets that we study and compare with some quite related different classes. This allows us to introduce and compare the corresponding linear operators.

Interpolating sequences on uniform algebras

Abstract We consider the problem of whether a given interpolating sequence for a uniform algebra yields linear interpolation. A positive answer is obtained when we deal with dual uniform algebras. Further we prove that if the Carleson generalized condition is sufficient for a sequence to be interpolating on the algebra of bounded analytic functions on the unit ball of c 0 , then it is sufficient for any dual uniform algebra.

Holomorphically ultrabornological spaces and holomorphic inductive limits

Abstract The holomorphically ultrabornological spaces are introduced. Their relation with other holomorphically significant classes of locally convex spaces is established and separating examples are given. Some apparently new properties of holomorphically barrelled spaces are included and holomorphically ultrabornological spaces are utilized in a problem posed by Nachbin.

Boundaries for algebras of analytic functions on function module Banach spaces

We consider the uniform algebra of continuous and bounded functions that are analytic on the interior of the closed unit ball of a complex Banach function module X. We focus on norming subsets of , i.e., boundaries, for such algebra. In particular, if X is a dual complex Banach space whose centralizer is infinite-dimensional, then the intersection of all closed boundaries is empty. This also holds in case that X is an -sum of infinitely many Banach spaces and further, the torus is a boundary.

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.

Interpolating sequences for bounded analytic functions

. We prove that any sequence in the open ball of a complex Banach space E, even in that of E**, whose norms are an interpolating sequence for H∞, is interpolating for the space of all bounded analytic functions on BE-The construction made yields that the interpolating functions depend linearly on the interpolated values.

The Composition Operation on Spaces of Holomorphic Mappings

AbstractWe discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.

Gleason Parts and Weakly Compact Homomorphismsbetween Uniform Banach Algebras

If points in nontrivial Gleason parts of a uniform Banach algebra have unique representing measures, then the weak and the norm topology coincide on the spectrum. We derive from this several consequences about weakly compact homomorphisms and discuss the case of other uniform Banach algebras arising in complex infinite dimensional analysis.

Spectra and essential spectral radii of composition operators on weighted Banach spaces of analytic functions

AbstractWe determine the spectra of composition operators acting on weighted Banach spaces Hv∞ of analytic functions on the unit disc defined for a radial weight v, when the symbol of the operator has a fixed point in the open unit disc. We also investigate in this case the growth rate of the Koenigs eigenfunction and its relation with the essential spectral radius of the composition operator.

Some algebras of symmetric analytic functions and their spectra

AbstractIn the spectrum of the algebra of symmetric analytic functions of bounded type on ℓp, 1 ≤ p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the ‘radius’ function. It is also proved that the algebra of analytic functions of bounded type on ℓ1 is isometrically isomorphic to an algebra of symmetric analytic functions on a polydisc of ℓ1. We also consider the existence of algebraic projections between algebras of symmetric polynomials and the corresponding subspace of subsymmetric polynomials.

The algebra of symmetric analytic functions on L∞

We consider the algebra of holomorphic functions on L∞ that are symmetric, i.e. that are invariant under composition of the variable with any measure-preserving bijection of [0, 1]. Its spectrum is identified with the collection of scalar sequences such that is bounded and turns to be separable. All this follows from our main result that the subalgebra of symmetric polynomials on L∞ has a natural algebraic basis.

Composition operators on uniform algebras, essential norms, and hyperbolically bounded sets

Let A be a uniform algebra, and let o be a self-map of the spectrum M A of A that induces a composition operator C o on A. The object of this paper is to relate the notion of "hyperbolic boundedness" introduced by the authors in 2004 to the essential spectrum of C o . It is shown that the essential spectral radius of C o , is strictly less than 1 if and only if the image of M A under some iterate o n of o is hyperbolically bounded. The set of composition operators is partitioned into "hyperbolic vicinities" that are clopen with respect to the essential operator norm. This partition is related to the analogous partition with respect to the uniform operator norm.

Essential norm estimates for composition operators on BMOA

Abstract We provide two function-theoretic estimates for the essential norm of a composition operator C φ acting on the space BMOA; one in terms of the n-th power φ n of the symbol φ and one which involves the Nevanlinna counting function. We also show that if the symbol φ is univalent, then the essential norm of C φ is comparable to its essential norm on the Bloch space.

Integral holomorphic functions

We define the class of integral holomorphic functions over Banach spaces; these are functions admitting an integral representation akin to the Cauchy integral formula, and are related to integral polynomials. After studying various properties of these functions, Banach and Frechet spaces of integral holomorphic functions are defined, and several aspects investigated: duality, Taylor series approximation, biduality and reflexivity. In this paper we define and study a class of holomorphic functions over infinite- dimensional Banach spaces admitting integral representation. Our purpose, and the motivation for our definition, are two-fold: we wish to obtain an integral repre- sentation formula …

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.

Polynomials generated by linear operators

We study the class of Banach algebra-valued n n -homogeneous polynomials generated by the n t h n^{th} powers of linear operators. We compare it with the finite type polynomials. We introduce a topology w E F w_{EF} on E , E, similar to the weak topology, to clarify the features of these polynomials.

Symmetric and finitely symmetric polynomials on the spaces ℓ∞ and L∞[0,+∞)

We consider on the space l∞ polynomials that are invariant regarding permutations of the sequence variable or regarding finite permutations. Accordingly, they are trivial or factor through c0. The analogous study, with analogous results, is carried out on L∞[0,+∞), replacing the permutations of N by measurable bijections of [0,+∞) that preserve the Lebesgue measure.

UNIQUENESS OF THE EXTENSION OF 2-HOMOGENEOUS POLYNOMIALS

Linearization of holomorphic mappings on fully nuclear spaces with a basis

In [13] Mazet proved the following result.If U is an open subset of a locally convex space E then there exists a complete locally convex space (U) and a holomorphic mapping δU: U→(U) such that for any complete locally convex space F and any f ɛ ℋ (U;F), the space of holomorphic mappings from U to F, there exists a unique linear mapping Tf: (U)→F such that the following diagram commutes;The space (U) is unique up to a linear topological isomorphism. Previously, similar but less general constructions, have been considered by Ryan [16] and Schottenloher [17].

The convolution operation on the spectra of algebras of symmetric analytic functions

Abstract We show that the spectrum of the algebra of bounded symmetric analytic functions on l p , 1 ≤ p + ∞ with the symmetric convolution operation is a commutative semigroup with the cancellation law for which we discuss the existence of inverses. For p = 1 , a representation of the spectrum in terms of entire functions of exponential type is obtained which allows us to determine the invertible elements.

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

WEAKLY COMPACT HOMOMORPHISMS BETWEEN SMALL ALGEBRAS OF ANALYTIC FUNCTIONS

The weak compactness of the composition operator CΦ(f) = f ○ Φ acting on the uniform algebra of analytic uniformly continuous functions on the unit ball of a Banach space with the approximation property is characterized in terms of Φ. The relationship between weak compactness and compactness of these composition operators and general homomorphisms is also discussed.

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…

Holomorphic Mappings of Bounded Type on (DF)-Spaces

We study the holomorphic functions of bounded type defined on (DF)-spaces. We prove that they are of uniformly bounded type. The space of all these functions is a Frechet space with its natural topology. Some consequences and related results are obtained.

Factorization of homomorphisms through H∞(D)

AbstractWeakly compact homomorphisms between (URM) algebras with connected maximal ideal space are shown to factor through H∞(D) by means of composition operators and to be strongly nuclear. The spectrum of such homomorphisms is also described. Strongly nuclear composition operators between algebras of bounded analytic functions are characterized. The path connected components of the space of endomorphisms on H∞(D) in the uniform operator topology are determined.

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) .

Bloch functions on the unit ball of an infinite dimensional Hilbert space

The Bloch space has been studied on the open unit disk of C and some ho- mogeneous domains of C n . We dene Bloch functions on the open unit ball of a Hilbert space E and prove that the corresponding space B(BE) is invariant under composition with the automorphisms of the ball, leading to a norm that- modulo the constant functions - is automorphism invariant as well. All bounded analytic functions on BE are also Bloch functions. ones, resulting the fact that if for a given n; the restrictions of the function to the n-dimensional subspaces have their Bloch norms uniformly bounded, then the function is a Bloch one and conversely. We also introduce an equivalent norm forB(BE) obtained by repla…