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

## TL;DR

This paper provides new, simplified proofs of the operator monotony of the square root function and the inverse function for positive operators, enhancing understanding of their properties in operator theory.

## Contribution

It introduces straightforward proofs for the monotonicity of the square root and inverse functions of operators, simplifying existing complex arguments.

## Key findings

- Proof that if 0 ≤ A ≤ B, then √A ≤ √B
- Proof that invertibility of A implies invertibility of B and B^{-1} ≤ A^{-1}
- Simplified methods for establishing operator monotony properties

## 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 $B^{-1}\leq A^{-1}$.