{"paper":{"title":"Schatten p-norm inequalities related to an extended operator parallelogram law","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Kichi-Suke Saito, Masaru Tominaga, Mohammad Sal Moslehian","submitted_at":"2011-06-15T19:10:00Z","abstract_excerpt":"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 $p=2$. M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.3057","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}