This is an announcement for the paper "Schatten p-norm inequalities related to a characterization of inner product spaces" by O. Hirzallah, F. Kittaneh, and M. S. Moslehian.

Abstract: Let $A_1, \cdots A_n$ be operators acting on a separable complex Hilbert space such that $\sum_{i=1}^n A_i=0$. It is shown that if $A_1, \cdots A_n$ belong to a Schatten $p$-class, for some $p>0$, then \begin{equation*} 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 \end{equation*} for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. Moreover, \begin{equation*} \sum_{i,j=1}^n|A_i\pm A_j|^2_p \leq 2n^{2/p} \sum_{i=1}^n |A_i|^2_p \end{equation*} for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. These inequalities are related to a characterization of inner product spaces due to E.R. Lorch.

Archive classification: math.OA math.FA

Mathematics Subject Classification: 46C15, 47A30, 47B10, 47B15

Remarks: 6 pages

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Submitted from: moslehian@ferdowsi.um.ac.ir

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