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

Abstract

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