Topologically transitive skew-products of backward shift operators and hypercyclicity

In this article we look at skew-products of multiples of the backward shift and examine conditions under which the skew-product is topologically transitive or hypercyclic in the second coordinate. We also give an application of the theory to iterated function systems of multiples of backward shift operators.

     

Interpolation by universal,hypercyclic functions

In the present article we study an interpolation problem for classes of analytic functions, in a systematic manner. More precisely, we provide sufficient conditions so that proper and “big”, in the Baire category sense, subclasses of analytic functions have an interpolation property at an infinite set of points. We then apply our main theorems to several classes of universal, hypercyclic functions.     

Hypercyclic semigroups and somewhere dense orbitsSemigroupes hypercycliques et orbites quelque part denses

We study hypercyclicity of linear strongly continuous semigroups. In the case of iterations of a single operator Bourdon and Feldman have recently proved that the existence of somewhere dense orbits implies hypercyclicity. We show the corresponding result for semigroups. As a consequence, a conjecture of Herrero concerning iterations of a single operator also holds for strongly continuous semigroups. To cite this article: G. Costakis, A. Peris, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 895–898.     

Topologically mixing hypercyclic operators

Let be a separable Fréchet space. We prove that a linear operator satisfying a special case of the Hypercyclicity Criterion is topologically mixing, i.e. for any given open sets there exists a positive integer such that for any We also characterize those weighted backward shift operators that are topologically mixing.

     

On the Radial Behavior of Universal Taylor Series

In this work we deal with universal Taylor series in the open unit disk, in the sense of Nestoridis; see [12]. Such series are not (C,k) summable at every boundary point for every k; see [7], [11]. In the opposite direction, using approximation theorems of Arakeljan and Nersesjan we prove that universal Taylor series can be Abel summable at some points of the unit circle; these points can form any closed nowhere dense subset of the unit circle.     

Cyclic operators with finite support

We begin a systematic study of cyclic operators with finite support, a notion which was introduced by P. Enflo in the paper where he solved the invariant subspace problem. We give spectral properties of these operators as well as surprising examples.     

On the Range of Universal Functions

We examine the range of universal Taylor series. We prove thatevery universal Taylor series on the unit disc assumes everycomplex number, with one possible exception, infinitely often.On the other hand, we prove that on any simply connected domainthere exist universal functions that omit one value. 2000 MathematicsSubject Classification 30B10.     

Electromagnetic scattering by a perfectly conducting obstacle in a homogeneous chiral environment: solvability and low‐frequency theory

The scattering of plane time‐harmonic electromagnetic waves propagating in a homogeneous isotropic chiral environment by a bounded perfectly conducting obstacle is studied. The unique solvability of the arising exterior boundary value problem is established by a boundary integral method. Integral representations of the total exterior field, as well as of the left and right electric far‐field patterns are derived. A low‐frequency theory for the approximation of the solution to the above problem, and the derivation of the far‐field patterns is also presented. Copyright © 2002 John Wiley & Sons, Ltd.     

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over R or C, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on Rn is n+1, thus complementing a recent result due to Feldman.     

J-Class Weighted Shifts on the Space of Bounded Sequences of Complex Numbers

We provide a characterization of J-class and J ^mix-class unilateral weighted shifts on in terms of their weight sequences. In contrast to the previously mentioned result we show that a bilateral weighted shift on cannot be a J-class operator. During this research the second author was fully supported by SFB 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik" at the University of Bielefeld, Germany. He would also like to express his gratitude to Professor H. Abels for his support.