Definition
Operator
An operator is a function that acts on elements of a structured space (typically a vector space or a space of functions) and returns elements of (usually) the same or another structured space.
Formally, an operator is just a map
where are spaces with some algebraic/analytic structure (vector space, inner product space, function space, etc.).
Linear Algebra
In linear algebra and functional analysis, “operator” often specifically means a linear map
between vector spaces, and especially an endomorphism