Clarkson-McCarthy inequalities with unitary and isometry orbits

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 vertical bar A + B/2 vertical bar(p) U* + V vertical bar A - B/2 vertical bar(p) V* <= vertical bar A vertical bar(p) + vertical bar B vertical bar(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, vertical bar A + B vertical bar(2) circle plus vertical bar A - B vertical bar(2) = U-0(vertical bar A vertical bar(2) + vertical bar B vertical bar(2))U-0* + V-0(vertical bar A vertical bar(2) + vertical bar B vertical bar(2))V-0* for some pair of 2n-by-n isometry matrices U-0, V-0. (C) 2020 Elsevier Inc. All rights reserved.