On The Absolute Value of The Product and the Sum of Linear Operators

Abstract

Let $A,B\in B(H)$. In the present paper, we establish simple and interesting facts on when we have $|A||B|=|B||A|$, $|AB|=|A||B|$, $|A\pm B|\le |A|+|B|$, $||A|-|B||\le |A\pm B|$ and $\Vert |A|-|B|\Vert \le \Vert A\pm B\Vert$, where $|\cdot |$ denotes the absolute value (or modulus) of an operator. The results give some other interesting consequences.

11 pages