Operators

overview

Summary

Operators are rules or mappings that take one or more inputs from a set (often a vector space or a space of functions) and produce an output in a set, frequently the same set. They generalize familiar actions like addition, multiplication, differentiation, and matrix transformation. Studying operators involves understanding their domains and codomains, algebraic and analytic properties (such as linearity, boundedness, and invertibility), and how they compose, represent, and act on structures. Operators are foundational across algebra, analysis, and applied mathematics, linking ideas like eigenvalues, spectra, and transforms.