# Positive operator

**Definition:** Given a Hilbert space H and *A* ∈ *L*(H), *A* is said to be a **positive operator** if ⟨*A**x*, *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.

Last modified:

Monday, October 26, 2015 - 17:56