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For $|\\gamma|<1$ operators $U_\\gamma$ are contractions with one-dimensional defects.\n  Restricting our attention on the non-trivial part of perturbation we assume that $b$ is cyclic for $U$. Then the operator $U_\\gamma$, $|\\gamma|<1$ is a completely non-unitary contraction, and thus un"},"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":"1308.3298","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-08-15T03:35:42Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"1ddf780f3511bc153ac43b06e1d262a19dbc857f98d19d104efe8cf936535937","abstract_canon_sha256":"70d3e5ee3694e3966bfb0c1b24a20fd89173d676f9bedea180196249d5b6fe23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:08.103406Z","signature_b64":"3ZWsq8jSvIZ6IghLtq9Ky/IeCbvQn7Ar+tv4acHYk88cHRNHrtxz+ty/5be5vE2QqLQRIcVKiqT9oqNDac7YBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c88c0f84aa94e5bd2d0b8dc6b98d94f04953c4d5eb7acef7dfda9763d2689d0","last_reissued_at":"2026-05-18T00:42:08.102848Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:08.102848Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Clark model in general situation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.FA","authors_text":"Constanze Liaw, Sergei Treil","submitted_at":"2013-08-15T03:35:42Z","abstract_excerpt":"For a unitary operator the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter $\\gamma, |\\gamma|=1$. Namely all such unitary perturbations are $U_\\gamma:=U+(\\gamma-1) (., b_1)_{\\mathcal H} b$, where $b\\in\\mathcal H, \\|b\\|=1, b_1=U^{-1} b, |\\gamma|=1$. For $|\\gamma|<1$ operators $U_\\gamma$ are contractions with one-dimensional defects.\n  Restricting our attention on the non-trivial part of perturbation we assume that $b$ is cyclic for $U$. 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