Definition: In mathematics, an operator is a function, that operates on (or modifies) another function.
In the following, though, we will consider the use that is made of operators in Quantum Mechanics and therefore focus upon particular classes of operators or operators with certain properties.
In the absence of superselection rules the following two axioms hold for any quantum system:
Axiom 1: The states of any physical system are in one-to-one correspondence with the vectors of an abstract, separable, infinite dimensional Hilbert space H.
Axiom 2: Every dynamical variable A of a quantum system can be represented by an operator A in the Hilbert space of the vector states, in the sense that the mean values of any dynamical variable function of A can be computed as the mean values of the same function of the operator A.
In particular, A must be a linear operator so that its mean values do not depend on the length of the vector chosen to compute them; whereas A must be a self-adjoint operator to guarantee the reality, and thus the physical consistency, of its mean values.