{"paper":{"title":"Operator Inequalities in $\\Phi$-Product Tensor Algebras: Invariance and Transform Sensitivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Michael K. Ng, Shih-Yu Chang","submitted_at":"2026-06-19T18:17:48Z","abstract_excerpt":"We study classical operator inequalities in $\\Phi$-product tensor algebras (a transformation $\\Phi$-based generalization of the $t$-product framework) for third-order tensors. Although these algebras are algebraically isomorphic under different unitary transforms $\\Phi$, we show that their quantitative behavior is not invariant. We prove that fundamental inequalities, including Golden--Thompson, Jensen, Klein, and Lieb, extend to the $\\Phi$-product setting with the same constants as in the matrix case. However, the associated defect -- the slack between the two sides of the inequality -- depen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21667","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.21667/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}