Loewner ordering

Let $H$ be a Hilbert space, and let $X,Y\in \mathrm{Sym}(E)$ be symmetric operators on $H$.

We define the Loewner order on $\mathrm{Sym}(E)$ by declaring if $X-Y$ is a positive semidefinite invertible bounded operator on $H$, and if $X-Y$ is a positive semidefinite invertible bounded operator on $H$.