On the solution of adjoint operator equations on hilbert spaces

This thesis consists of eight chapters. In the first chapter, general information about the thesis and a brief history of the relevant subject are given. In the second chapter, literature review is given. In the third chapter, some basic definitions and theorems that will be used in other chapters of the thesis are given. In the fourth chapter, properties of selfandjoint operators are discussed and characterizations of self-adjoint operators among symmetric operators by Von Neumann are developed. In the fifth chapter, in our study on adjoints of sums and products of operators in Hilbert spaces, we develop the perturbation theory of selfadjoint and essentially selfadjoint operators originating from Nelson, Kato, Rellich and Wüst by providing sufficient and necessary conditions that guarantee the equations (𝐴 +𝐵)∗ = (𝐴∗)+(𝐵∗) ve (𝐴𝐵)∗ = 𝐵∗𝐴∗ for densely defined unbounded operators A and B. In the sixth chapter, adjoints of Hilbert space operators are studied. In the seventh chapter, the solutions of nonlinear operator equations in the form of A𝑥 +𝜆𝑇𝑥 = 𝑓 are examined. In these equations, A and T are nonlinear operators defined in Hilbert space and λ is a parameter. In the study, the solutions of these equations are made using Banach, Mann and Ishikawa iteration methods.