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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.

MSC:
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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