# Loewner ordering

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

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

Title Loewner ordering LoewnerOrdering 2013-03-22 14:40:15 2013-03-22 14:40:15 mathcam (2727) mathcam (2727) 4 mathcam (2727) Definition msc 40A30 Loewner order