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

Abstract

We prove that for all $0\le\alpha\le 2/3$$$\Vert |A|^{\alpha}X-X|B|^{\alpha}\Vert \le2^{2-\alpha}\Vert X\Vert^{1-\alpha} \Vert AX-XB\Vert^{\alpha},$$for all bounded Hilbert space operators $A=A^*$, $B=B^*$ and $X$, as well as$$\Vert |A|^{\alpha}-|B|^{\alpha}\Vert \le2^{2-\alpha} \Vert A-B\Vert^{\alpha},$$for arbitrary bounded $A$ and $B$.