Schatten p-norm inequalities related to an extended operator parallelogram law

Let $\mathcal{C}_p$ be the Schatten $p$-class for $p>0$. Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: If $\mathbf{A}=\{A_1,A_2,...,A_n\}$ and $\mathbf{B}=\{B_1,B_2,...,B_n\}$ are two sets of operators in $\mathcal{C}_2$, then $$\sum_{i,j=1}^n\|A_i-A_j\|_2^2 + \sum_{i,j=1}^n\|B_i-B_j\|_2^2 = 2\sum_{i,j=1}^n\|A_i-B_j\|_2^2 - 2\Norm{\sum_{i=1}^n(A_i-B_i)}_2^2.$$ In this paper, we give generalizations of this as pairs of inequalities for Schatten $p$-norms, which hold for certain values of $p$ and reduce to the equality above for $p=2$. Moreover, we present some related inequalities for three sets of operators.

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