Operator

linear-algebra

Definition

Operator

An operator is a function that acts on elements of a structured space and returns elements of a structured space.

Formally, an operator is a map

$$T:X\to Y,$$

where $X$ and $Y$ carry some algebraic or analytic structure, such as a vector space, an inner product space, or a function space.

A functional is a special kind of operator whose domain is a space of functions and whose codomain is usually a scalar field.

For operators, one usually writes

$$Tx\equiv T(x)$$

Linear Algebra

In linear algebra and functional analysis, “operator” often specifically means a linear map

$$T:V\to W$$

between vector spaces, and especially an endomorphism

$$T:V\to V.$$