In mathematics, “operator” has two main definitions. 1) An operator is a mathematical symbol, for example +, that represents an operation or action. In this case, the operator + says to add. The entire mathematical statement might be: 2 + 3. The operator + says: add 2 and 3. Other familiar operators are – (subtract) and X (multiply).
Operator in calculus. In calculus, d/dx is an operator that says to find the derivative of a function. For example, d/dx f(x) means to find the derivative of the function f(x). Calculus includes other operators as well.
2) Operator in vector mathematics.* In vector mathematics, an operator is a mathematical expression that says to take one vector and manipulate it so as to create another vector of a particular kind. This is also called “mapping” one vector to another. An operator or “mapping” is like a mathematical function: feed it one vector as an input, and, then, output another vector. This type of operator can be used by physicists to describe how one physical state, represented by a vector, becomes another, represented by another vector.
*Vector mathematics is also called “linear algebra.” While arithmetic manipulates individual numbers (3, 1017, etc.), vector math manipulates groups of numbers, for example the two-number group 〈0, 1〉 or the three-number group 〈1,0,3〉.