# Positive operator

Definition: Given a Hilbert space  H and  A ∈ L(H),  A is said to be a positive operator if  ⟨Ax, x⟩ ≥ 0 for every  x ∈ H.

A positive operator on a complex Hilbert space is necessarily a symmetric operator and has a self-adjoint extension that is also a positive operator.

The set of positive bounded operators on a Hilbert space forms a cone in the algebra of all bounded operators.