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. Operators can be thought of as generalizations of matrices, acting on infinite-dimensional spaces, which makes them essential in areas like quantum mechanics and signal processing. One key concept in operator theory is the spectrum of an operator, which consists of the set of values that describe its behavior. This includes eigenvalues, which are special values that indicate how an operator transforms certain functions. Operator theory also connects to other fields, such as Hilbert spaces and Banach spaces, providing a framework for solving differential equations and studying functional equations.