Superposition Operator on Fock and Harmonic Fock Spaces

Eshetu, Yonas

dc.contributor.author	Eshetu, Yonas
dc.date.accessioned	2025-07-28T12:22:54Z
dc.date.available	2025-07-28T12:22:54Z
dc.date.issued	2025-05
dc.identifier.uri	http://ir.bdu.edu.et/handle/123456789/16798
dc.description.abstract	The superposition operator is a nonlinear operator that arises in a wide range of mathematical contexts, particularly in functional analysis and operator theory. It plays a crucial role in several areas including nonlinear functional analysis, integral equations, differential equations, and harmonic analysis. The study of this operator has a long history especially in the context of real-valued func tion domains. In recent years, there has been increasing interest in exploring its properties within spaces of analytic functions. Much of this research fo cuses on identifying the various forms of the operator that map one space to another, as well as analyzing its boundedness, compactness, and continuity an alytic structures. However, its topological, dynamical, and spectral structures in these spaces remain less understood. This dissertation aims to address this gap by thoroughly investigating these aspects of the operator on Fock and harmonic Fock spaces. The dissertation is structured into six chapters with each chapter focusing on specific structural aspects of the operator. The first chapter provides historical context, introduces the operator, and presents the essential background results that lay the foundation for the subsequent analysis. Chapter two presents our findings on the various topological properties of weighted superposition operator on analytic Fock spaces. More specifically, we analyze key structures such as boundedness from below, strict singularity, or der boundedness, compact difference, Fréchet differentiability, Hilbert adjoint, self-adjointness, closed range, fixed point, and the invariant subspace problem. Additionally, we characterize the operator’s global homeomorphism property in terms of its ray invertibility conditions. The results in this chapter effectively illustrate how the weighted superposition operator serves as a prototypical ex amplefor highlighting the fundamental differences between linear and nonlinear operator theories. The dissertation then delves into Chapter three where the iterated structures of the operator are studied. We establish that the operator is power bounded if and only if it is mean ergodic. We further show the Fock spaces do not support cyclic weighted superposition operators, implying that no orbit of the operator forms a frame. We also identify conditions under which the operator preserves frames, tight frames, Riesz bases, and semigroup structures. In particular, the results show that the operator preserves frames if and only if it is linear. vi Chapter four focuses on the spectral properties of the operator on Fock spaces. We follow several approaches in nonlinear spectral theory and iden tify its various spectral sets. The results show that most of the spectral forms introduced so far coincide and contain singletons for this operator. The classi cal, asymptotic, and connected eigenvalues, and some numerical ranges of the operator are also identified. We further prove the operator is both linear and odd asymptotically with respect to the pointwise multiplication operator on the spaces. Expanding the scope of the underlying spaces, Chapter five reexamines the topological and spectral properties of the superposition operator on Harmonic Fock spaces Fp H . It is proven that these spaces do not support a nontrivial com pactness structure for the operator, and that a nontrivial order-bounded structure exists only when the operator acts on F∞ H . Furthermore, we show every su perposition operator in these spaces is a closed map and provide an explicit expression for its range. Unlike in the analytic Fock spaces setting, where no nontrivial bounded below superposition operator exists, we identify conditions under which the operator exhibits bounded-below behavior in harmonic Fock spaces. Additionally, we establish local invertibility conditions that imply global invertibility and observe that the operator’s spectral sets and numerical ranges are broader in harmonic Fock spaces compared to Fock spaces. Finally, Chapter six concludes the dissertation by summarizing the main f indings in tabular form and discussing potential directions for future research	en_US
dc.language.iso	en	en_US
dc.subject	Mathematics	en_US
dc.title	Superposition Operator on Fock and Harmonic Fock Spaces	en_US
dc.type	Thesis	en_US