operator

is: a symbolic instruction to carry out some (mathematical) action on a function. [P10.4, P11.3]

can range: from simple multiplication by a number, to quite complicated acts such as taking repeated derivatives with respect to a given variable. [P10.4, P11.3]

may correspond: in quantum mechanics, to an observable such as momentum or kinetic energy, (in which particular cases the operator is a differential operator). [P10.4, P11.3]