Topological And Dynamical Structures Of Composition Operators On Generalized Fock Spaces

In this thesis we study various properties of composition operators acting betweenrngeneralized Fock spaces Fp'rnand Fq'rnwith weight functions ' grow faster thanrnthe classical Gaussian weight function 1rn2 jzj2 and satisfy some mild smoothnessrnconditions. Let be an analytic map on the complex plane. Then for p 6= q, wernhave shown that the composition operator C : Fp'rn! Fq'rnis bounded if and onlyrnif it is compact. This result shows a signi_cance di_erence with the analogousrnresult for the case when C acts between the classical Fock spaces or generalizedrnFock spaces where the weight functions grow slower than the Gaussian function.rnWe further study some topological structure of composition operators on thernspaces. It is shown that the di_erence of two composition operators is compact ifrnand only if both are compact, and hence cancellation phenomenon fails to exist.rnWhile each non-compact bounded composition operator is an isolated point, thernset of all compact composition operators forms a connected components of thernspace of the operators under the operator norm topology. Moreover, SchattenrnSp(F2'rn) class membership, spectra, hyponormality of the composition operatorsrnare characterized.rnWe also study various dynamical structure of composition operators C on therngeneralized Fock spaces Fp'rnand the weighted composition operatorsW(u; ) de_nedrnon the classical Fock spaces Fp. It is shown that all composition operators onrnthe spaces are power bounded. Several conditions characterizing uniformly meanrnergodic composition operators are provided. We have identi_ed operators W(u; )rnthat are power bounded and uniformly mean ergodic on the spaces, and thesernproperties are described in terms of easy to apply conditions which are basedrnmerely on the values ju(0)j and ju( brn1À€€a )j, where the numbers a and b are fromrnlinear expansion of the symbol (z) = az + b. We have proved that compositionrnoperators C and weighted composition operatorsW(u; ) can not be supercyclic onrntheir respective Fock spaces. Furthermore, the operator C ; (z) = az+b; jaj _rn1 and b = 0 when jaj = 1, is cyclic if and only if an 6= a, while W(u; ) is cyclic ifrnand only if the corresponding composition operator is cyclic and u fails to vanishrnon C. The set of periodic points of C is also determined. Conditions under whichrnthe operators satisfy the Ritt's resolvent growth conditions are also identi_ed. Inrnparticular, we show that a non-trivial composition operator on the Fock spacesrnsatis_es such growth condition if and only if it is compact.

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