An operator is just a function which maps one vector into another. In the case when our vectors are functions in function space, an operator takes one function into another. For example, the derivative is an operator which maps a function into its derivative. Linear operators are a special class of operators which are very important in quantum mechanics. A linear operator acting on the sum of two vectors is the same as the sum of the operator acting on the each vector individually.

If an operator maps a vector |phi1> into a vector |phi2>, then the adjoint of the operator takes <phi1| into <phi2;. Hermitian operators are those operators for which the adjoint is the same as the original operator. Observables like the energy, momentum, and position are all hermitian operators.