On a Theorem of Godefroy and Shapiro

G. Godefroy and J. H. Shapiro have shown that every operator on , that commutes with all translation operators , and that is not a scalar multiple of the identity is hypercyclic. We show that they are even frequently hypercyclic. In addition, we obtain growth conditions that may be satisfied by corresponding frequently hypercyclic entire functions.     

On Non-Distinguished Matrix Domains

In this note matrices A and B are presented such that the matrix domains (?_∞) _A and (?₁)_B are not distinguished. We also give an example of a matrix domain c _A whose bidual is not distinguished.     

Universal functions for composition operators with non-automorphic symbol

For sequences (φ _n ) of eventually injective holomorphic self-maps of planar domains Ω, we present necessary and sufficient conditions for the existence of holomorphic functions f on whose orbits under the action of (φ _n ) are dense in H (Ω). It is deduced that finitely connected, but non-simply connected domains never admit such universal functions. On the other hand, if we allow arbitrary sequences of holomorphic self-maps (φ _n ), the situation changes dramatically.     

Strong Weighted Mean Summability and Kuttner's Theorem

In this paper we study sequence spaces that arise from the conceptof strong weighted mean summability. Let q = (q_n) be a sequenceof positive terms and set Q_n = ⁿ_k=1q_k. Then the weighted meanmatrix M_q = (a_nk) is defined by if kn, a_nk=0 if k>n. It is well known that M_q defines a regular summability methodif and only if Q_n. Passing to strong summability, we let 0<p<.Then , are the spaces of all sequences that are strongly M_q-summablewith index p to 0, strongly M_q-summable with index p and stronglyM_q-bounded with index p, respectively. The most important specialcase is obtained by taking M_q = C₁, the Cesàro matrix,which leads to the familiar sequence spaces w₀(p), w(p) and w(p), respectively, see [4, 21]. We remark that strong summabilitywas first studied by Hardy and Littlewood [8] in 1913 when theyapplied strong Cesàro summability of index 1 and 2 toFourier series; orthogonal series have remained the main areaof application for strong summability. See [32, 6] for furtherreferences. When we abstract from the needs of summability theory certainfeatures of the above sequence spaces become irrelevant; forinstance, the q_k simply constitute a diagonal transform. Hence,from a sequence space theoretic point of view we are led tostudy the spaces     

Strongly omnipresent integral operators

An operatorT on the spaceH(G) of holomorphic functions on a domainG is strongly omnipresent whenever there is a residual set of functionsfH(G) such thatT f exhibits an extremely wild behaviour near the boundary. The concept of strong omnipresence was recently introduced by the first two authors. In this paper it is proved that a large class of integral operators including Volterra operators with or without a perturbation by differential operators has this property, completing earlier work about differential and antidifferential operators.The work of the first two authors has been partially supported by DGES grant PB96-1348 and the Junta de Andalucía.     

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 Bermúdez, Bonilla and Martinón. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Fréchet space. This complements recent results due to Bès 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.

     

Frequently dense orbits

We study the notion of frequent hypercyclicity that was recently introduced by Bayart and Grivaux. We show that frequently hypercyclic operators satisfy the Hypercyclicity Criterion, answering a question of Bayart and Grivaux [Trans. Amer. Math. Soc., in press]. We also disprove a conjecture therein concerning frequently hypercyclic weighted shifts, and we prove that vectors which have a somewhere frequently dense orbit are frequently hypercyclic. To cite this article: K.-G. Grosse-Erdmann, A. Peris, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Frequently hypercyclic subspaces

We study the existence of frequently hypercyclic subspaces for a given operator, that is, the existence of closed infinite-dimensional subspaces in which every non-zero vector is frequently hypercyclic. We attack the problem with any of the three methods that have been used for hypercyclic subspaces: a constructive approach, an approach via left-multiplication operators, and an approach via tensor products.