A Dunkl-Williams type inequality for absolute value operators
The Dunkl-Williams inequality states that for any nonzero elements \(x,y\) in a normed linear space \(X\) the following inequality holds: \[ \left\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\right\| \leq \frac{4\|x-y\|}{\|x||+\|y\|}. \] The authors generalize this inequality considering the algebra \(B({\mathcal H})\) of all bounded linear operators acting on a complex Hilbert space \({\mathcal(H)}\). The main result is the following: Let \(A,B\in B({\mathcal H})\) be operators with polar decompositions \(A=U|A|\) and \(B=V|B|\), and let \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\). Then \[ |(U-V)|A||^2\leq p|A-B|^2+q(|A|-|B|)^2. \] The equality holds iff \(p(A-B)=qV(|B|-|A|)\) and \(U^*U=V^*V.\) Also, they give a refinement of the equality conditions.
scientific article; zbMATH DE number 5706981
scientific article