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 $A=A^*$, $B=B^*$ 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 $B$.

Keywords: Singular values, three line theorem for operators, unitarily invariant norms.

MSC: 47A30, 47B05, 47B10, 47B15

Pages:  151--152     

Volume  49 ,  Issue  3$-$4 ,  1997