José Bonet

COMPOSITION OPERATORS BETWEEN WEIGHTED BANACH SPACES OF ANALYTIC FUNCTIONS

We characterize those analytic self-maps ’ of the unit disc which generate bounded or compact composition operators C’ between given weighted Banach spaces H 1 v or H 0 v of analytic functions with the weighted sup-norms. We characterize also those composition operators which are bounded or compact with respect to all reasonable weights v.

DIFFERENCES OF COMPOSITION OPERATORS BETWEEN WEIGHTED BANACH SPACES OF HOLOMORPHIC FUNCTIONS

We consider differences of composition operators between given weighted Banach spaces H∞ v or H 0 v of analytic functions with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. Some examples illustrate our results. MSC 2000: 47B33, 47B38.

Dual Density Conditions in (DF)— spaces, I

We define the two “dual density conditions” (DDC) and (SDDC) for locally convex topological vector spaces and study them in the setting of the class of (DF)- spaces (originally introduced by A. Grothendieck [14]). We show that for a (DF)- space E, (DDC) is equivalent to the metrizability of the bounded subsets of E, and prove that such a space E has (DDC) resp. (SDDC) if and only if the space l∞(E) of all bounded sequences in E is quasibarrelled resp. bornological.As a consequence, we can then characterize the barrelled spaces % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

HYPERCYCLIC AND CHAOTIC CONVOLUTION OPERATORS

{\cal L}_b(\lambda_1,\ E)

Transitive and Hypercyclic Operators on Locally Convex Spaces

of continuous linear mappings from a Köthe echelon space λ1into a locally complete (DF)- space E; for purposes of a comparison, we also provide the corresponding characterization of the quasibarrelled resp. bornological (DF)- tensor products (λ1)b′ ⊗εE. Our results on the (DF)- spaces of type% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Linear Chaos on Fréchet Spaces

{\cal L}_b(\lambda_1,\ E)

Real Analytic Curves in Fréchet Spaces and Their Duals.

and (λ1)b′) ⊗εE are of special interest in view of the recent negative solution, due to J. Taskinen (see [25]), of Grothendieck’s “problème des topologies” ([15]). — In part II of the article, we will treat weighted inductive limits of spaces of continuous functions and their projective hulls (cf. [6]) as an application.In his study of ultrapowers of locally convex spaces, S. Heinrich [16] had found it necessary to introduce the “density condition”. Our article [2] investigated this condition, mainly in the setting of Fréchet spaces, and with applications to distinguished echelon spaces λ1. However, on the way to the main theorems of [2], it became apparent that the “right” setting for most of this material was a dual reformulation of the density condition in the context of (DF)- spaces, and this observation prompted the present research.

On Mean Ergodic Operators

Every convolution operator on a space of ultradifferentiable functions of Beurling or Roumieu type and on the corresponding space of ultradistributions is hypercyclic and chaotic when it is not a multiple of the identity. The operator of differentiation is hypercyclic on the space A −∞ , but it need not be hypercyclic on radial weighted algebras of entire functions.

Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Frechet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space ϕ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on ϕ. (2) The space of all test functions for distributions, which is also a complete direct sum of Frechet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Frechet space contains a dense hyperplane that admits no transitive operator. 2000 Mathematics Subject Classification 47A16 (primary), 46A03, 46A04, 46A13, 37D45 (secondary).

Weighted (LF)‐spaces of Continuous Functions

This is a survey on recent results about hypercyclicity and chaos of continuous linear operators between complete metrizable locally convex spaces. The emphasis is put on certain contributions from the authors, and related theorems.