UNIVERSALITY OF COMPOSITION OPERATOR WITH CONFORMAL MAP ON THE UPPER HALF PLANE

Abstract

The main theme of this dissertation is the dynamical behavior of composition operators on the Frechet space H(P) of holomorphic functions on the upper half-plane. In this dissertation, we prove a new version of the Seidel and Walsh Theorem for the Frechet space H(P). Indeed, we obtain a necessary and sufficient condition for the sequence of linear fractional transformations such that the sequence of composition operators induced by this sequence of LFT's for the Frechet space H(P) is universal. For that, we use the Riemann Mapping Theorem to transfer dynamical results on the space H(D) of holomorphic functions on D to the space of holomorphic functions H(P). Furthermore, we generalize our first result by proving equivalent conditions for a sequence of composition operators in the space H(D) to be universal. Consequently, taking the point of view that hypercyclicity is a special case of universality, we obtain a new criterion for a linear fractional transformation so that the composition operator that induced by this LFT is hypercyclic on H(P). Indeed, we provide necessary and sufficient conditions in terms of the coefficients a, b, c, d of a linear fractional transformation defined on P so that the composition operator that induced by this LFT is hypercyclic on H(P). Moreover, we use this result to derive a necessary and sufficient condition so that the composition operator that defined on H(D) is hypercyclic on H(D). Motivated by the Denjoy-Wolff Theorem, we further work on the conformal map of the upper half-plane P is to make a connection between the hypercyclicity and the limit of the iterations of the conformal map of the upper half-plane P. In particular, we give a complete characterization for the limit point of LFT in the extended boundary of the upper half plane. Similarly, we provide an analogous result for the unit disk D. Finally, we obtain a new universal criterion in the space of holomorphic functions on a bounded simply connected region G that is not the whole complex plane. We show that a sequence of composition operators on space of holomorphic functions on a bounded simply connected region G is universal if and only if there are a boundary limit point w and a subsequence such that this subsequence is convergence uniformly on compact subsets of G. Our last result extends a result of Grosse-Erdmann, and Manguillot in a particular case when the complement G has a nonempty interior.