Löwner order

For Hermitian operators $A$ and $B$, one writes $A\le B$ when $B-A$ is positive semidefinite. This partial order is also called the semidefinite order.

For bounded positive self-adjoint operators, $0\le A\le B$ implies $A^{1/2}\le B^{1/2}$. Thus the nonnegative square-root function is operator monotone.