Jochen Wengenroth

Hypercyclic operators on non-locally convex spaces

We transfer a number of fundamental results about hypercyclic operators on locally convex spaces (due to Ansari, Bes, Bourdon, Costakis, Feldman, and Peris) to the non-locally convex situation. This answers a problem posed by A. Peris [Multi-hypercyclic operators are hypercyclic, Math. Z. 236 (2001), 779-786].

Transitive and Hypercyclic Operators on Locally Convex Spaces

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).

Prescribed Derivatives of Holomorphic Functions

Let z 0 be a point in an open set

Weighted (LF)-spaces of measurable functions

G \subseteq {\shadC}

Whitney extension operators without loss of derivatives

and

Examples on projective spectra of (LB)-spaces

\Lambda \subseteq {\shadN}_0

Strong duals of projective limits of (LB)-spaces

an infinite set. We study the problem when it is possible to find for all prescribed derivatives f x of order x k v (satisfying the obvious bounds implied by the radius of convergence for the maximal disc around z 0 in G ) an analytic function f on G with

5. The derived functors of Hom

f^{(\nu )} (z_0) / \nu ! = \alpha _\nu

4. Uncountable projective spectra

for all x k v In that case, v is called G -interpolating (in z 0 ). We prove by functional analytic methods (a variation of the Banach-Schauder open mapping theorem and Köthes description of the dual of the Fréchet space H(G) ) that this property only depends on the intersection of G with the maximal circle around z 0 in G This enables us to characterize G -interpolating sets v by a condition on the density of v if, for instance, the intersection of G with the maximal circle around z 0 is an arc.

A conjecture of Palamodov about the functors Extk in the category of locally convex spaces

We provide a study for a class of (LF)-spaces where the steps are projective limits of weighted Köthe function spaces. For this class we characterize several regularity conditions in terms of the weights and give a projective description of the inductive limit topology similar as it was done by Bierstedt and Bonet for weighted (LF)-spaces of continuous functions. This extends the work of Reiher on the corresponding (LB)-spaces and generalizes results of Vogt about (LF)-sequence spaces.