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The class of the bounded operators on $L^2(T^n)$ with analytic orbit under the action of $T^n$ by conjugation with the translation operators is shown to coincide with the class of the zero-order pseudodifferential operators on $T^n$ whose discrete symbol $(a_j)_{j\\in Z^n}$ is uniformly analytic, in the sense that there exists $C>1$ such that the derivatives of $a_j$ satisfy $|\\partial^\\alpha a_j(x)|\\leq C^{1+|\\alpha|}\\alpha!$ for all $x\\in T^n$, all $j\\in Z^n$ and all $\\alpha\\in N^n$. This implies that this class of pseudodifferential operators is a sp"},"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":"1610.06439","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-19T19:38:14Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"f56713f2d8e870f838c627301dd53195f6df05d1d5b769dcaf56504bbd63118a","abstract_canon_sha256":"3485a22e2b11bb742110bf8f43b645bf5ff7054b2bbc6b73b031569fc499ccac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:43.193431Z","signature_b64":"HTzpcDQXNhv9XBG+NbsXXdoJrH6Hb7Ws3/AnTISStFlVC0sK1/gfieug0AaYBQ/74cS85BUNMoFRLPmsYy6GAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eee6a411b42c42113748b812c55e283504d366436e27a8f5e8d009873d0e0a09","last_reissued_at":"2026-05-18T01:01:43.192762Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:43.192762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Operators with analytic orbit for the torus action","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Rodrigo A. H. M. Cabral, Severino T. Melo","submitted_at":"2016-10-19T19:38:14Z","abstract_excerpt":"Let $T^n$ denote the n-dimensional torus. The class of the bounded operators on $L^2(T^n)$ with analytic orbit under the action of $T^n$ by conjugation with the translation operators is shown to coincide with the class of the zero-order pseudodifferential operators on $T^n$ whose discrete symbol $(a_j)_{j\\in Z^n}$ is uniformly analytic, in the sense that there exists $C>1$ such that the derivatives of $a_j$ satisfy $|\\partial^\\alpha a_j(x)|\\leq C^{1+|\\alpha|}\\alpha!$ for all $x\\in T^n$, all $j\\in Z^n$ and all $\\alpha\\in N^n$. 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