Closed Range and Dynamical Structures of  Volterra-type Integral Operators on Fock Spaces

Asnake, Zenaw

dc.contributor.author	Asnake, Zenaw
dc.date.accessioned	2025-07-28T12:16:49Z
dc.date.available	2025-07-28T12:16:49Z
dc.date.issued	2025-05
dc.identifier.uri	http://ir.bdu.edu.et/handle/123456789/16796
dc.description.abstract	Integral operators are fundamental tools for modeling various real-world problems and hold a central position in operator theory. This dissertation ex plores the intricate properties of these operators, with a specific focus on gen eralized Volterra-type integral operators V(g,ψ) and J(g,ψ). While substantial research has been conducted on properties such as boundedness, compactness, and essential norm, other fundamental properties such as order boundedness, closed range, spectral characteristics, and dynamical structures remains insuffi ciently explored. This dissertation aims to fill that gap by extensively exploring these properties within the framework of Fock spaces. The dissertation is organized into six chapters, each addressing a specific aspect of the operators on Fock spaces. Chapter 1 provides a comprehensive review of the relevant literature on integral operators and presents some prelim inary results that serves as the foundation for subsequent investigations. Chapter 2 addresses the closed range problem for generalized Volterra-type integral operators on Fock spaces FP. We first provide a complete solution to this problem by using several key concepts which include sampling sets, reverse Fock–Carleson measures, the reproducing kernel thesis, boundedness from be low, and the essential boundedness of two relevant functions M(g,ψ) and M(g,ψ). Building on these results, we solve the problem in terms of a simple condition which only requires to verify whether the coefficient of the first-degree term in the Taylor expansion of the function ψ lies on the unit circle. Additionally, we prove that no nontrivial closed range integral operators exist between two distinct Fock spaces. Explicit expressions for their ranges are also derived, re vealing that these ranges consist solely of functions in the space that vanish at the origin. Chapter 3 focuses on the spectra and iterated structures of the integral op erators. To identify spectral sets, we first establish an interesting connection between the spectra of the integral operators and weighted composition opera tors acting on appropriate spaces. It is shown that while the operator V(g,ψ) does not admit eigenvalues, all eigenvalues of J(g,ψ) are of the form g(z0)am, where a is from the expansion ψ(z) = az+b and z0 is the fixed point of ψ. As an application of these spectral results, the iterates of the operators are analyzed, v and their power bounded structures are characterized. Chapter 4 investigates the cyclicity and frame structures of the operators on Fock spaces. We demonstrate that Fock spaces do not support supercyclic or convex-cyclic generalized Volterra-type integral operators, nor do they support cyclic structures for the operator J(g,ψ). Interestingly, we show that J(g,ψ) admits a cyclic and convex-cyclic structures when its action is restricted to its closed range F0 p, and hence falls under a subspace convex-cyclic category of operators. We further explore the dynamical sampling properties from the perspective of frame theory. It is shown that no element in the space has a dense orbit under the action of the integral operators. A loosely related question was when the op erators preserve frame structures in the Fock space F2. It is found that while the operators themselves fail to preserve such structures, their adjoints do preserve under some interesting conditions. The order bounded structures of the integral operators are studied in Chapter 5. We characterize the structures in terms of an Lp integrability condition of the functions M(g,ψ) and M(g,ψ). The proof of the results are based on the action of the operators on the normalized reproducing kernel functions. Finally, in Chapter 6, we summarizes the dissertation’s main findings and discuss potential directions for future research. By addressing these less ex plored but fundamental aspects of integral operators, this dissertation contributes to a deeper understanding of their properties , particularly in the context of Fock spaces.	en_US
dc.language.iso	en	en_US
dc.subject	Mathematics	en_US
dc.title	Closed Range and Dynamical Structures of Volterra-type Integral Operators on Fock Spaces	en_US
dc.type	Thesis	en_US