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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 $\mathbb{Z}$.