Properties and Analysis of a Second Order Differential Operator with Normality and Orthogonality

This research paper investigates the normality and eigenvalue problems associated with second-order differential operators. The study explores the properties and applications of these operators in the field of functional analysis. The main results show that the second-order differential operator under consideration is normal, demonstrating its adherence to the fundamental property of normality. The orthogonality of the eigenspaces corresponding to distinct eigenvalues, providing insights into the spectral properties of the operator is also established. Additionally, the relationship between the null spaces of the operator and its higher powers is shown, shedding light on the behavior of the operator under repeated application. The ndings contribute to the understanding of differential operators and their role in various mathematical contexts.