Norm inequality for the class of self-adjoint absolute value generalized derivations

Abstract

We prove that for all $0\le \alpha \le 2/3$$$\Vert |A{|}^{\alpha }X-X|B{|}^{\alpha }\Vert \le 2^{2-\alpha }\Vert X{\Vert }^{1-\alpha }\Vert AX-XB{\Vert }^{\alpha },$$for all bounded Hilbert space operators $A=A^{\ast }$, $B=B^{\ast }$ and $X$, as well as$$\Vert |A{|}^{\alpha }-|B{|}^{\alpha }\Vert \le 2^{2-\alpha }\Vert A-B{\Vert }^{\alpha },$$for arbitrary bounded $A$ and $B$.