Homomorphisms and composition operators on algebras of analytic functions of bounded type

Let U and V be convex and balanced open subsets of the Banach spaces X and Y, respectively. In this paper we study the following question: given two Fréchet algebras of holomorphic functions of bounded type on U and V, respectively, that are algebra isomorphic, can we deduce that X and Y (or X^* and Y^*) are isomorphic? We prove that if X^* or Y^* has the approximation property and H_wu(U) and H_wu(V) are topologically algebra isomorphic, then X^* and Y^* are isomorphic (the converse being true when U and V are the whole space). We get analogous results for H_b(U) and H_b(V), giving conditions under which an algebra isomorphism between H_b(X) and H_b(Y) is equivalent to an isomorphism between X^* and Y^*. We also obtain characterizations of different algebra homomorphisms as composition operators, study the structure of the spectrum of the algebras under consideration and show the existence of homomorphisms on H_b(X) with pathological behaviors.     

Algebras of holomorphic mappings in (DF)-spaces

Let U be an absolutely convex open subset of a complex barrelled (DF)-space E and let F be a commutative Banach algebra with identity. Let H_b(U, F) be the space of holomorphic mappings from U into F that are bounded on the U-bounded sets and let H_b (U, F) be the space of the holomorphic mappings from U into F that are uniformly weakly continuous on the U-bounded sets, both endowed with the topology τ_b of uniform convergence on the U-bounded sets. The spectra of (H_wu (U, F), τ_b) and (H_b(U, F), τ_b) are studied.     

On a Bornological Structure in Infinite-Dimensional Holomorphy

The bornology (b) of bounded subsets with respect to continuous convergence is used on spaces of holomorphic functions. It is shown that Hom_coH_b(U) ? U for a circled convex open subset U of a complete nuclear space. Exponential laws for spaces of holomorphic functions with bornological structures are proved and the connection with Colombeau's Silva holomorphic functions is established.     

Gantmacher Type Theorems for Holomorphic Mappings

Given a holomorphic mapping of bounded type g ∈ H_b(U, F), where U ? E is a balanced open subset, and E, F are complex Banach spaces, let A : H_b(F) ∈ H_b(U) be the homomorphism defined by A(f) = fog for all f ∈ H_b(F). We prove that: (a) for F having the Dunford-Pettis property, A is weakly compact if and only if g is weakly compact; (b) A is completely continuous if and only if g(W) is a Dunford-Pettis set for every U-bounded subset W ? U. To obtain these results, we prove that the class of Dunford - Pettis sets is stable under projecti ve tensor products. Moreover, we diaracterize the reflexivity of the space H_b(U,F) and prove that E' and F have the Schur property if and only if H_b(U, F) has the Schur property. As an application, we obtain some results on linearization of holomorphic mappings.     

The approximation property for spaces of holomorphic functions on infinite-dimensional spaces I

For an open subset U of a locally convex space E, let (H(U),τ₀) denote the vector space of all holomorphic functions on U, with the compact-open topology. If E is a separable Fréchet space with the bounded approximation property, or if E is a (DFC)-space with the approximation property, we show that (H(U),τ₀) has the approximation property for every open subset U of E. These theorems extend classical results of Aron and Schottenloher. As applications of these approximation theorems we characterize the spectra of certain topological algebras of holomorphic mappings with values in a Banach algebra.     

On the distinguished character of the function spaces of holomorphic mappings of bounded type

Given two complex normed spaces E and F, F complete, and a balanced open subset U of E, we prove that the space $\text{H}$_(b(U, F) of the holomorphic mappings f: U → F of bounded type, endowed with its natural topology τ_b, is a distinguished quasi-normable Fréchet space, which is not a Schwartz space unless dim E < ∞ and dim F < ∞.     

The spectrum of an algebra of weakly continuous holomorphic mappings

Let U be an arbitrary absolutely convex open subset of a complex Banach space E and let F be a Banach algebra with identity. The spectrum of the algebra H_b(U, F) of the holomorphic mappings from U and F which are bounded on the U-bonded subsets of U is studied in case E′ has the approximation properly.     

Compact and Weakly Compact Homomorphisms on Fréchet Algebras of Holomorphic Functions

We study homomorphisms between Fréchet 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 H_b(V), V absolutely convex open set.     

The approximation property for spaces of holomorphic functions on infinite dimensional spaces II

Let H(U) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τω and τδ respectively denote the compact-ported topology and the bornological topology on H(U). We show that if E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on E is weakly continuous on bounded sets, then (H(U),τω) and (H(U),τδ) have the approximation property for every open subset U of E. The classical space c₀, the original Tsirelson space T^∗ and the Tsirelson^∗-James space are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains.     

Clark model in the general situation

For a unitary operator U in a Hilbert space H the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter γ, ?γ? = 1. Namely, all such unitary perturbations are operators U_γ:= U + (γ ? 1)( ·, b₁)_Hb, where b ∈ H, ∥b∥ = 1, b₁ = U^?1b, ?γ? = 1. For ?γ? < 1, the operators U_γ are contractions with one-dimensional defects.     

A Banach-Dieudonné theorem for germs of holomorphic functions

Let $\text{H}$(U) denote the space of all holomorphic functions on an open subset U of a complex Fréchet space E. Let $\text{H}$(K) denote the space of all holomorphic germs on a compact subset K of E. It is shown that $\text{H}$(K), with a natural topology, is the inductive limit of a suitable sequence of compact subsets, within the category of all topological spaces. As an application of this result it is shown that the compact-ported topology introduced by Nachbin coincides with the compact-open topology on $\text{H}$(U) whenever U is a balanced open subset of a Fréchet-Schwartz space. This last result improves earlier results of P. Boland and S. Dineen [Bull. Soc. Math. France106 (1978), 311–336], R. Meise [Proc. Roy. Irish Acad. Sect. A81 (1981), 217–223], and others.     

Metrizability of spaces of holomorphic functions

In this paper we prove that if U is an open subset of a metrizable locally convex space E of infinite dimension, the space H(U) of all holomorphic functions on U, endowed with the Nachbin-Coeuré topology τδ, is not metrizable. Our result can be applied to get that, for all usual topologies, H(U) is metrizable if and only if E has finite dimension.     

Isometries of non-self-adjoint operator algebras

We study isometries of certain non-self-adjoint operator algebras by means of the structure of the complete holomorphic vector fields on their unit balls and the associated partial Jordan triple products. We show that isometries of nest sub-algebras of B(H) are of the form T UTW or T UJT^*JW, where U, W are suitable unitary operators and J a fixed involution of H.     

Weakly and Separately Holomorphic Functions on Compact Determining Polydiscs in (DFN)-Spaces

The aim of this paper is to find some necessary and sufficient conditions by means of the property (DN) of E and the property (LB _∞) of F under which H _w (X, F) = H(X, F) and H _δ (X, H _b (F*)) = H(X, H _b (F*)) where E, F are Fréchet spaces and X is a compact determining polydisc in E*. At the same time, we also study the problem of the local Dirichlet representation of separately holomorphic functions.     

Jordan zero-product preserving additive maps on operator algebras

Let Φ:A→B be an additive surjective map between some operator algebras such that AB+BA=0 implies Φ(A)Φ(B)+Φ(B)Φ(A)=0. We show that, under some mild conditions, Φ is a Jordan homomorphism multiplied by a central element. Such operator algebras include von Neumann algebras, C^∗-algebras and standard operator algebras, etc. Particularly, if H and K are infinite-dimensional (real or complex) Hilbert spaces and A=B(H) and B=B(K), then there exists a nonzero scalar c and an invertible linear or conjugate-linear operator U:H→K such that either Φ(A)=cUAU⁻¹ for all A∈B(H), or Φ(A)=cUA^∗U⁻¹ for all A∈B(H).     

Extensions of quantum enveloping algebras

In this paper, we define a class of extended quantum enveloping algebras U _q (f(K, J)) and some new Hopf algebras, which are certain extensions of quantum enveloping algebras by a Hopf algebra H. This construction generalizes some well-known extensions of quantum enveloping algebras by a Hopf algebra and provides a large of new noncommutative and noncocommutative Hopf algebras.     

On the theorem of hartogs in non-Archimedean valued fields

Let U = U₀ × U₁ × … × U_n be an open polyring in a non-Archimedean valued, locally non-compact field. Let the function f be defined in the polyring U and satisfy the following conditions: (1) f is holomorphic for every x ∈ U₀ separately in each of the rest variables y_i ∈ U_i, i = 1, 2,…,n; (2) f is holomorphic in x ∈ U₀ for every (y₁,…,y_n) ∈ V₁ × … × V_n, where V_i is a certain disk from the ring U_i. Then, if the valuation is dense, the function f is holomorphic in the polyring U. If the valuation is discrete, then the function f is holomorphic in a domain close to the polyring U.     

Vector spaces of non-extendable holomorphic functions

In this paper, we analyze the linear structure of the family H _e (G) of holomorphic functions on a domain G of the complex plane that are not analytically continuable beyond the boundary of G. We prove that H _e (G) contains, except for zero, a dense algebra; and, under appropriate conditions, the subfamily of H _e (G) consisting of boundary-regular functions contains dense vector spaces with maximal dimension as well as infinite dimensional closed vector spaces and large algebras. We also consider the case in which G is a domain of existence in a complex Banach space. The results obtained complete or extend a number of previous results by several authors.