Journal of Operator Theory

Volume 55, Issue 1, Winter 2006 pp. 135-151.

Supercyclic and hypercyclic non-convolution operators

Authors: Henrik Petersson
Author institution: School of Mathematical Sciences, Chalmers Goeteborg University, Goeteborg, SE-412 96, Sweden

Summary: A continuous linear operator

$T:X\to X$ is hypercyclic/super\-cyclic if there is a vector

$f\in X$ such that the orbit

$\orb (T,f) =\{ T^n f\}$ /respec\-tively the set of scalar-multiples of the orbit elements, forms a dense set. A famous theorem, due to G.\ Godefroy \& J.\ Shapiro, states that every non-scalar convolution operator, on the space

$\Hb$ of entire functions in

$d$ variables, is hypercyclic (and thus supercyclic). This motivates us to study cyclicity of operators on

$\Hb$ outside the set of convolution operators. We establish large classes of supercyclic and hypercyclic non-convolution operators.

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