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

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
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References:
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