Operator Theory is a branch of functional analysis that studies linear operators on function spaces. It focuses on understanding how these operators behave, their properties, and their applications in various mathematical contexts. This theory is essential for solving differential equations and understanding quantum mechanics, where operators represent physical observables. In Operator Theory, concepts such as Hilbert spaces and Banach spaces are crucial. These spaces provide the framework for analyzing operators, including bounded and unbounded types. The study of spectral theory within Operator Theory helps in understanding the spectrum of an operator, which is vital for determining its behavior and stability.