Nonnegative (positive-semidefinite) operator

Let $H$ be an inner product space and let $A:H\to H$ be a self-adjoint linear operator.

The operator $A$ is nonnegative (or positive-semidefinite) if

\langle Ax,x\rangle\ge 0\quad \text{for all }x\in H.

Examples

If $A=B^\ast B$ for some linear operator $B$ , then $\langle Ax,x\rangle=\langle Bx,Bx\rangle\ge 0$ , so $A$ is nonnegative.

Remarks

Context. Nonnegative operators correspond to convex quadratic forms. In finite dimensions, this matches the usual matrix notion of positive semidefiniteness.