Synonym of mapping and function. Often used to refer to mappings where the domain and codomain are, in some sense a space of functions.

Examples: differential operator, convolution operator.

In the study of algebraic systems such as groups and rings, an operator often refers to a mapping from some cartesian power $A^{\lambda}$ of a set $A$ to the set $A$ where $\lambda$ is a cardinal. For example, multiplication of integers can be thought of as an operator from $\mathbb{Z}^2$ to $\mathbb{Z}$