Weighted Composition Operators and Supercyclicity Criterion

We consider an equivalent condition to the property of Supercyclicity Criterion, and then we investigate this property for the adjoint of weighted composition operators acting on Hilbert spaces of analytic functions.

Supercyclicity and Hypercyclicity of an Isometry Plus a Nilpotent

Suppose thatXis a separable normed space and the operatorsAandQare bounded onX. In this paper, it is shown that ifAQ=QA,Ais an isometry, andQis a nilpotent then the operatorA+Qis neither supercyclic nor weakly hypercyclic. Moreover, if the underlying space is a Hilbert space andAis a co-isometric operator, then we give sufficient conditions under which the operatorA+Qsatisfies the supercyclicity criterion.

On hypercyclicity and supercyclicity criteria

We show that the Hypercyclicity Criterion coincides with other existing hypercyclicity criteria and prove that a wide class of hypercyclic operators satisfy the Criterion. The results obtained extend or improve earlier work of several authors. We also unify the different versions of the Supercyclicity Criterion and show that operators with dense generalised kernel and dense range are supercyclic.