Operator
High school
An operator is a function that takes a function as its input and gives a function as its output.
Undergraduate (functional analysis)
An operator is a map from a vector space to a vector space, usually linear and often between spaces of functions.
Examples
Non-examples
Things that may be mistaken for examples.
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$f(x) = x^2$ on $\reals$
High school
$f(x) = x^2$ on $\reals$ is not an operator, because its inputs are numbers, not elements of a space of functions.
Notes
In elementary arithmetic, “operator” means the symbol for an operation, such as $+$ or $\times$. That is a different meaning.