Some inequalities for commutators and an application to spectral variation. (English) Zbl 0752.47005

Bhatia, Rajendra; Davis, Chandler; Kittaneh, Fuad

Aequationes Math. 41, No. 1, 70-78 (1991).

Let \(A,B\) be two Hermitian or two unitary operators on a complex Hilbert space and let \(\Gamma\geq\gamma I\) (\(\gamma>0\)) be a positive operator. The authors prove that for every unitarily invariant norm one has \(||| A\Gamma-\Gamma B|||>\gamma||| A- B|||\).
It is known that when \(|||\cdot|||=\|\cdot\|_ 2\), the Hilbert- Schmidt norm, this inequality remains true for any two normal operators \(A,B\). The authors conjecture that the inequality is true in general for normal \(A,B\). They also provide interesting corollaries, remarks and examples.

Reviewer: K.N.Boyadzhiev (Ada)

MSC:

47A63	Linear operator inequalities
47B47	Commutators, derivations, elementary operators, etc.
15A42	Inequalities involving eigenvalues and eigenvectors
15B57	Hermitian, skew-Hermitian, and related matrices
15A60	Norms of matrices, numerical range, applications of functional analysis to matrix theory
65F99	Numerical linear algebra

Keywords:

unitarily invariant norm; Hilbert-Schmidt norm

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