Operator Theory

Operator Theory is a branch of mathematics and functional analysis that deals with linear operators in Hilbert space and their properties. It is mainly used in mathematical physics, signal processing, quantum mechanics, noncommutative algebra, system theory, and numerical analysis. Operator Theory is important in obtaining solutions to linear and differential equations, as well as in analyzing partial differential equations. Operator Theory is also important for the development of new techniques for signal analysis, for instance, for filtering, for extracting features from data, for signal and image understanding. Operator Theory is also used in signal processing for spectral analysis, signal de-noising, joint time-frequency and time-scale signal processing, and for wavelet analysis. It is also used for control theory, in which it is used to study stability, controllability and observability of linear and nonlinear systems. Operator Theory is also used for network analysis and synthesis, for instance for computing the scattering parameters.