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We show that $A\\leq_u B$ if and only if $f(g(A)^r)\\leq_uf(g(B)^r)$ for any increasing operator convex function $f$, any operator monotone function $g$ and any positive number $r$. We present some sufficient conditions under which if $B\\leq A\\leq U^*BU$, then $B=A=U^*BU$. Finally we prove that if $A^n\\leq U^\\ast A^nU$ for all $n\\in\\mathbb{N}$, then $A=U^\\ast AU$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1204.2222","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-04-10T17:12:24Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"bd01446eb1aa94b4b74f5b0e3dc4d23d9c455a8ca85d161c65a0133c6f9b5b0f","abstract_canon_sha256":"4bb6018fbc0aebaa4cf0fa08eb863749726ad05c1458783e8a1873d2f96e4b9e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:23.043146Z","signature_b64":"Q6CUmhNbjfh7US33ZGNFhgkNLNoNcPN8sAGLN1aFCf96ucaBIXfaqRL1RwjO/1Go1GMQBcOoC49bJ+e1GbgAAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbab3273e78de75f995ba8b17598005948ca1d5e3a6e3a6e5b4fefbdee7d7692","last_reissued_at":"2026-05-18T03:55:23.042615Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:23.042615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the binary relation $\\leq_u$ on self-adjoint Hilbert space operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.OA","authors_text":"H. 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