New proofs of the operator monotony of the square root and the inverse

Abstract

Let $A,B\in B(H)$. We present among others a simple proof of the widely known result stating that if $0\leq A\leq B$, then $\sqrt A\leq \sqrt B$. The same idea is used to prove that if $0\leq A\leq B$ and $A$ is invertible, then $B$ too is invertible and $B^{-1}\leq A^{-1}$.