Let A1, ···An be operators acting on a separable complex Hilbert space such that ∑n i=1 Ai = 0. It is shown that if A1, ···An belong to a Schatten p-class, for some p > 0, then 2p/2np−1 n ∑ i=1Aip p  n ∑ i, j=1Ai ±Ajp p for 0 < p  2, and the reverse inequality holds for 2  p < ∞ . Moreover, n ∑ i, j=1Ai ±Aj2 p  2n2/p n ∑ i=1Ai2 p for 0 < p  2, and the reverse inequality holds for 2  p < ∞ . These inequalities are related to a characterization of inner product spaces due to E.R. Lorch.

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