A hypercyclicity criterion with applications

A continuous linear operator on a topological vector space X is called hypercyclic if there is x∈X such that the orbit {Tnx}_n?0 is dense in X. We establish a criterion for hypercyclicity, and study some applications. In particular, we establish hypercyclic left-multipliers on the space L(X,Y) of continuous linear operators between X and Y, provided with the topology of uniform convergence on bounded sets, for some spaces X,Y of holomorphic functions.     

Primitive elements in rings of continuous functions

Let be a surjective continuous map between compact Hausdorff spaces. The map π induces, by composition, an injective morphism C(Y)→C(X) between the corresponding rings of real-valued continuous functions, and this morphism allows us to consider C(Y) as a subring of C(X). This paper deals with algebraic properties of the ring extension C(Y)⊆C(X) in relation to topological properties of the map . We prove that if the extension C(Y)⊆C(X) has a primitive element, i.e., C(X)=C(Y)[f], then it is a finite extension and, consequently, the map π is locally injective. Moreover, for each primitive element f we consider the ideal and prove that, for a connected space Y, If is a principal ideal if and only if is a trivial covering.