A remark on normal derivations. (English) Zbl 0894.47003

Duggal, B. P.

Proc. Am. Math. Soc. 126, No. 7, 2047-2052 (1998).

Summary: Given a Hilbert space \(H\), let \(A,S\) be operators on \(H\). Anderson has proved that if \(A\) is normal and \(AS=SA\), then \(\| AX-XA+S\| \geq\| S\| \) for all operators \(X\). Using this inequality, Du Hong-Ke has recently shown that if (instead) \(ASA=S\), then \(\| AXA-X+S\| \geq\| A\| ^{-2}\| S\| \) for all operators \(X\). In this note we improve the Du Hong-Ke inequality to \(\| AXA-X+S\| \geq\| S\| \) for all operators \(X\). Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.

MSC:

47A30	Norms (inequalities, more than one norm, etc.) of linear operators
47A63	Linear operator inequalities
47B15	Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B48	Linear operators on Banach algebras

Keywords:

normal derivation; norm inequality; contraction; unitarily invariant norms; Du Hong-Ke inequality; Anderson inequalities