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

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