Non-Weakly Supercyclic Weighted Composition Operators

We give sufficient conditions under which a weighted composition operator on a Hilbert space of analytic functions is not weakly supercyclic. Also, we give some necessary and sufficient conditions for hypercyclicity and supercyclicity of weighted composition operators on the space of analytic functions on the open unit disc. 1. Introduction Let denote the open unit disc in the complex plane. Let be a Hilbert space of analytic functions defined on such that , and for each in , the linear functional of point evaluation at given by is bounded. By a Hilbert space of analytic functions we mean the one satisfying the above conditions. For , let denote the linear functional of point evaluation at on ; that is, for every in . Since is a bounded linear functional, the Riesz representation theorem states that for some A well-known example of a Hilbert space of analytic functions is the weighted Hardy space. Let be a sequence of positive numbers with The weighted Hardy space is defined as the space of functions analytic on such that Since the function is in , we see that These spaces are Hilbert spaces with the inner product for every and in Let for every nonnegative integer . Then is an orthogonal basis. As a particular consequence of this fact, the polynomials are dense in It is clear that . We call the set the standard basis for The sequence of weights allows us to consider the generating function which is analytic on Take . It is easy to see that for any function in , where ; moreover, The classical Hardy space, the Bergman space, and the Dirichlet space are weighted Hardy spaces with weights, respectively, given by , and . Weighted Bergman and Dirichlet spaces are also weighted Hardy spaces. Let be an automorphism of the disc. Recall that is elliptic if it has one fixed point in the disc and the other in the complement of the closed disc, hyperbolic if both of its fixed points are on the unit circle, and parabolic if it has one fixed point on the unit circle (of multiplicity two). Recall that a multiplier of is an analytic function on such that . The set of all multipliers of is denoted by . If is a multiplier, then the multiplication operator , defined by , is bounded on . Also, note that for each , . It is known that . In fact, since the constant functions are in , every function in is analytic on . Moreover, if , then which implies that for every , and so In what follows, suppose that and that is an analytic self-map of such that is in for every . An application of the closed graph theorem shows that the weighted composition operator defined by is bounded.