Clarkson-McCarthy inequalities with unitary and isometry orbits
Résumé
A refinement of a trace inequality of McCarthy establishing the uniform convexity of the Schatten p-classes for p > 2 is proved: if A, B are two n-by-n matrices, then there exists some pair of n-by-n unitary matrices U, V such that U A + B 2 p U * + V A − B 2 p V * ≤ |A| p + |B| p 2. A similar statement holds for compact Hilbert space operators. Another improvement of McCarthy's inequality is given via the new operator parallelogramm law, |A + B| 2 ⊕ |A − B| 2 = U 0 (|A| 2 + |B| 2)U * 0 + V 0 (|A| 2 + |B| 2)V * 0 for some pair of 2n-by-n isometry matrices U 0 , V 0 .
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