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Schatten \(p\)-norm inequalities related to a characterization of inner product spaces. (English) Zbl 1201.47013
From the authors’ abstract: “Let \(A_1,\dots,A_n\) be operators acting on a separable complex Hilbert space such that \(\sum_{i=1}^n A_i=0\). It is shown that if \(A_1,\dots,A_n\) belong to a Schatten \(p\)-class, for some \(p>0\), then
\[ 2^{p/2}n^{p-1} \sum_{i=1}^n \|A_i\|_p^p\leq \sum_{i,j=1}^n \|A_i\pm A_j\|_p^p \]
for \(0<p\leq 2\), and the reverse inequality holds for \(2\leq p<\infty\). Moreover,
\[ \sum_{i,j=1}^n \|A_i\pm A_j\|_p^2\leq 2n^{2/p} \sum_{i=1}^n \|A_i\|_p^2 \]
for \(0<p\leq 2\), and the reverse inequality holds for \(2\leq p<\infty\).”
These inequalities are special cases of more general results.

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
46C15 Characterizations of Hilbert spaces
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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