Topological and Dynamical Properties of Volterra-Type Integral and Differential Operators on Fock-Type Spaces

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Addis Ababa University


Many real world problems can be modelled using different forms of differential and integral operators. Due to this, studying various properties of these class of operators have been an important area of research in mathematics. The study has attracted lots of interest especially on function related operator theory in the past two decades. In this thesis, we consider these operators and study several topological and dynamical properties on Fock-type spaces. The thesis is organized into four chapters. In the first chapter we discuss some preliminary results on the Volterra-type integral operators. In particular, properties like boundedness and compactness of the operators on some spaces are reviewed. In the second chapter, we study various properties of the generalized Volterra-type integral and the generalized composition operators acting on the classical Fock spaces. We describe some properties in terms of growth and integrability conditions which are simpler to apply than those already known Berezin-type integral transform characterizations. In the third chapter we apply those simple conditions obtained in Chapter 2 and study some topological structures of the space of the operators under the operator norm. In particular, we prove that the difference of two nontrivial Volterra-type integral operators is compact if and only if both are compact. We also consider the structures on other Fock-type spaces with iii weight function growing faster than the classical Gaussian function. It is proved that while the space contains each noncompact operator as its isolated points, it fails to admit essentially isolated points. We have also shown that the space of all bounded Volterra companion and multiplication operators are path connected. In the last chapter, various dynamical properties of the Volterra-type integral, differentiation and Hardy operators on Fock-type spaces are studied. More specifically, properties like cyclicity, supercyclicity, hypercyclicity, power boundedness and uniformly mean ergodicity are characterized in terms of workable conditions. As an application, we prove that the differentiation operator satisfies the Ritt’s resolvent condition if and only if it is power bounded and uniformly mean ergodic, whereas the Hardy operator always satisfies such a condition



Topological and Dynamical, Properties of Volterra, Type Integral and Differential, Differential Operators on Fock-Type Spaces