Yigrem, Tesfa (PhD)Norway, Western (Professor)Seyoum, Werkaferahu2020-09-302023-11-092020-09-302023-11-092020-08-08http://10.90.10.223:4000/handle/123456789/22537In this thesis we study various properties of composition operators acting between generalized Fock spaces Fp' and Fq' with weight functions ' grow faster than the classical Gaussian weight function 1 2 jzj2 and satisfy some mild smoothness conditions. Let be an analytic map on the complex plane. Then for p 6= q, we have shown that the composition operator C : Fp' ! Fq' is bounded if and only if it is compact. This result shows a signi_cance di_erence with the analogous result for the case when C acts between the classical Fock spaces or generalized Fock spaces where the weight functions grow slower than the Gaussian function. We further study some topological structure of composition operators on the spaces. It is shown that the di_erence of two composition operators is compact if and only if both are compact, and hence cancellation phenomenon fails to exist. While each non-compact bounded composition operator is an isolated point, the set of all compact composition operators forms a connected components of the space of the operators under the operator norm topology. Moreover, Schatten Sp(F2' ) class membership, spectra, hyponormality of the composition operators are characterized. We also study various dynamical structure of composition operators C on the generalized Fock spaces Fp' and the weighted composition operatorsW(u; ) de_ned on the classical Fock spaces Fp. It is shown that all composition operators on the spaces are power bounded. Several conditions characterizing uniformly mean ergodic composition operators are provided. We have identi_ed operators W(u; ) that are power bounded and uniformly mean ergodic on the spaces, and these properties are described in terms of easy to apply conditions which are based merely on the values ju(0)j and ju( b 1􀀀a )j, where the numbers a and b are from linear expansion of the symbol (z) = az + b. We have proved that composition operators C and weighted composition operatorsW(u; ) can not be supercyclic on their respective Fock spaces. Furthermore, the operator C ; (z) = az+b; jaj _ 1 and b = 0 when jaj = 1, is cyclic if and only if an 6= a, while W(u; ) is cyclic if and only if the corresponding composition operator is cyclic and u fails to vanish on C. The set of periodic points of C is also determined. Conditions under which the operators satisfy the Ritt's resolvent growth conditions are also identi_ed. In particular, we show that a non-trivial composition operator on the Fock spaces satis_es such growth condition if and only if it is compact.enTopologicalDynamical Structures of CompositionOperators on GeneralizedSpacesThesis