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We show that either the range of $H$ consists of zero trace elements, or there is a scalar sequence $\\{\\lambda_n\\}$ and an invertible $S$ in $M_m$ such that $$ H(x) =\\sum_{n\\geq 1} \\lambda_n S^{-1}x^nS, \\quad\\forall x \\in M_m,%\\eqno{(\\ddag)} $$ or $$ H(x) =\\sum_{n\\geq 1} \\lambda_n S^{-1}(x^t)^nS, \\quad\\forall x \\in M_m. $$ Here, $x^t$ is the transpose of the matrix $x$. In the latter case, we alway"},"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":"1402.6849","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-02-27T10:20:10Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"c3e12b5815a4c8abe8e4d2aa76382239b3c09e4dfa09827f9de3dd30b85f46f2","abstract_canon_sha256":"f40fcfbd1c94ce1568bb3ab5e9c9e28d7bbebadef58b3c6fea4c3761ff6d7e85"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:36.807624Z","signature_b64":"+Vnf6VKTlA1oUY9KhhytsVYEdd1jHgBzasnkX/JjvQax3qS/DPUrD4V/5rf6qSp8jYEprqYT82OumpWkrfk0CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9348d1e3aea3f9eca0ebd284550fe9060a658f234f0ea553f07507e3ded662d","last_reissued_at":"2026-05-18T02:57:36.806851Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:36.806851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Chingjou Liao, Ngai-Ching Wong, Qingying Bu","submitted_at":"2014-02-27T10:20:10Z","abstract_excerpt":"Let $H:M_m\\to M_m$ be a holomorphic function of the algebra $M_m$ of complex $m\\times m$ matrices. Suppose that $H$ is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of $H$ consists of zero trace elements, or there is a scalar sequence $\\{\\lambda_n\\}$ and an invertible $S$ in $M_m$ such that $$ H(x) =\\sum_{n\\geq 1} \\lambda_n S^{-1}x^nS, \\quad\\forall x \\in M_m,%\\eqno{(\\ddag)} $$ or $$ H(x) =\\sum_{n\\geq 1} \\lambda_n S^{-1}(x^t)^nS, \\quad\\forall x \\in M_m. $$ Here, $x^t$ is the transpose of the matrix $x$. 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