Karl-Goswin Grosse-Erdmann

Universal families and hypercyclic operators

Part II. Specific Universal Families and Hypercyclic Operators 3. The real analysis setting 3a. Universal power and Taylor series 3b. Universal primitives 3c. Universal orthogonal series 3d. Universal series for convergence a.e. 3e. Further real universalities and hypercyclicities 4. The complex analysis setting 4a. Universal and hypercyclic composition operators 4b. Holomorphic monsters 4c. Hypercyclic differential operators 4d. Universal power and Taylor series 4e. Universal matrices 5. Hypercyclic operators in classical Banach spaces 6. A concrete universal object: the Riemann zeta-function

Frequently hypercyclic operators and vectors

We study frequently hypercyclic operators, a natural new concept in hypercyclicity that was recently introduced by F. Bayart and S. Grivaux. We derive a strengthened version of their Frequent Hypercyclicity Criterion, which allows us to obtain examples of frequently hypercyclic operators in a straightforward way. Moreover, Bayart and Grivaux have noted that the frequent hypercyclicity setting differs from general hypercyclicity in that the set of frequently hypercyclic vectors need not be residual. We show here that, under weak assumptions, this set is only of first category. Motivated by this we study the question of whether one may write every vector in the underlying space as the sum of two frequently hypercyclic vectors. This investigation leads us to the introduction of a new notion, that of Runge transitivity.

The blocking technique : weighted mean operators and Hardy's inequality

The blocking technique.- The sequence spaces c(a, p, q) and d(a, p, q).- Applications to matrix operators and inequalities.- Integral analogues.

On the universal functions of G.R. Maclane

An entire function f is called universal (in the sense of MacLane) if a suitable sequence f (nk) of derivatives of f converges to any preassigned entire function, locally uniformly in the whole plane. We show: Given any function there exists a universal function f such that , while there can be no universal function with .

Rate of growth of frequently hypercyclic functions

We study the rate of growth of entire functions that are frequently hypercyclic for the differentiation operator or the translation operator. Moreover, we prove the existence of frequently hypercyclic harmonic functions for the translation operator and we study the rate of growth of harmonic functions that are frequently hypercyclic for partial differentiation operators.

Regularity properties of functional equations and inequalities

SummaryBy a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line.We consider here a locally compact topological groupX with some Haar measure. Then the following generalizes Trautners theorem: Theorem.Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (zn)in W there is a subsequence (znk)and points y and xkin M with znk =xk ·y−1for k ∈ℕ. Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski. Theorem.Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X → R and H: R × X → T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X → T of f(x · y) = H(g)(x), y) for x, y∈X is continuous.Theorem.Let G: X × X → ℝ be a mapping. If there is a subset M of X of positive finite Haar measure such that for each y∈X the mapping x ↦ G(x, y) is bounded above on M, then any solution f: x → ℞ of f(x · y) ⩽ G(x, y) for x, y∈X is locally bounded above. We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument off.

Existence and nonexistence of hypercyclic semigroups

In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from-and considerably shorter than-the one recently given by Bermudez, Bonilla and Martinon. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Frechet space. This complements recent results due to Bes and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.

Weakly mixing operators on topological vector spaces

In this paper we review some known characterizations of the weak mixing property for operators on topological vector spaces, extend some of them, and obtain new ones.ResumenEn este artículo revisamos algunas caracterizaciones conocidas de la propiedad débil mezclante para operadores en espacios vectoriales topológicos, extendemos alguna de ellas, y obtenemos otras nuevas.

Upper frequent hypercyclicity and related notions

Enhancing a recent result of Bayart and Ruzsa we obtain a Birkhoff-type characterization of upper frequently hypercyclic operators and a corresponding Upper Frequent Hypercyclicity Criterion. As an application we characterize upper frequently hypercyclic weighted backward shifts on sequence spaces, which in turn allows us to come up with various counter-examples in linear dynamics that are substantially simpler than those previously obtained in the literature. More generally, we introduce the notion of upper Furstenberg families