{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:X2ORBG53MHPMCDWOGAROX4TEBQ","short_pith_number":"pith:X2ORBG53","schema_version":"1.0","canonical_sha256":"be9d109bbb61dec10ece3022ebf2640c04b1abdca5d6e3071d981fa963f929ba","source":{"kind":"arxiv","id":"2608.03574","version":1},"attestation_state":"computed","paper":{"title":"On the Dilation Theory and Canonical Decomposition of $\\mathbf{\\Theta}_n$-Contractions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.FA","authors_text":"Aparna Gupta, Avijit Pal, Bhaskar Paul","submitted_at":"2026-08-04T12:32:26Z","abstract_excerpt":"This paper studies the domain $\\mathbf{\\Theta}_n$ from the perspective of operator theory. We obtain several characterizations of $\\mathbf{\\Theta}_n$-contractions (respectively, $\\mathbf{\\Theta}_n$-unitaries and $\\mathbf{\\Theta}_n$-isometries) and establish their relationships with $\\Gamma_n$-contractions (respectively, $\\Gamma_n$-unitaries and $\\Gamma_n$-isometries), tetrablock contractions (respectively, tetrablock unitaries and tetrablock isometries), and $\\mathbf{\\Theta}_{n+1}$-contractions (respectively, $\\mathbf{\\Theta}_{n+1}$-unitaries and $\\mathbf{\\Theta}_{n+1}$-isometries). We prove t"},"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":"2608.03574","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-08-04T12:32:26Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"2e2332fa836bba15bfa71d15f1c01b795b1953bcdb044dc89bae927596c9dde7","abstract_canon_sha256":"642272833020fefb651c9ebc0badec66bed1ab88d5e7f69949dbf8f67ad8be06"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-05T01:36:40.491564Z","signature_b64":"7u896F2o1iQcEoPlFkKGYfgWR2UYBqhNtMS0fO0VXlN487mi6Pin2Io8AQRIkycfUbWc6kosC2R2aEkcD5PiAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be9d109bbb61dec10ece3022ebf2640c04b1abdca5d6e3071d981fa963f929ba","last_reissued_at":"2026-08-05T01:36:40.489981Z","signature_status":"signed_v1","first_computed_at":"2026-08-05T01:36:40.489981Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Dilation Theory and Canonical Decomposition of $\\mathbf{\\Theta}_n$-Contractions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.FA","authors_text":"Aparna Gupta, Avijit Pal, Bhaskar Paul","submitted_at":"2026-08-04T12:32:26Z","abstract_excerpt":"This paper studies the domain $\\mathbf{\\Theta}_n$ from the perspective of operator theory. We obtain several characterizations of $\\mathbf{\\Theta}_n$-contractions (respectively, $\\mathbf{\\Theta}_n$-unitaries and $\\mathbf{\\Theta}_n$-isometries) and establish their relationships with $\\Gamma_n$-contractions (respectively, $\\Gamma_n$-unitaries and $\\Gamma_n$-isometries), tetrablock contractions (respectively, tetrablock unitaries and tetrablock isometries), and $\\mathbf{\\Theta}_{n+1}$-contractions (respectively, $\\mathbf{\\Theta}_{n+1}$-unitaries and $\\mathbf{\\Theta}_{n+1}$-isometries). We prove t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.03574","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/2608.03574/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2608.03574","created_at":"2026-08-05T01:36:40.490446+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.03574v1","created_at":"2026-08-05T01:36:40.490446+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.03574","created_at":"2026-08-05T01:36:40.490446+00:00"},{"alias_kind":"pith_short_12","alias_value":"X2ORBG53MHPM","created_at":"2026-08-05T01:36:40.490446+00:00"},{"alias_kind":"pith_short_16","alias_value":"X2ORBG53MHPMCDWO","created_at":"2026-08-05T01:36:40.490446+00:00"},{"alias_kind":"pith_short_8","alias_value":"X2ORBG53","created_at":"2026-08-05T01:36:40.490446+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.13366","citing_title":"Dilation and Functional Models for Pure $\\mathbf{\\Theta}_n$-Contractions and the von Neumann Inequality on Distinguished Varieties in $\\mathbf{\\Theta}_n$","ref_index":23,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/X2ORBG53MHPMCDWOGAROX4TEBQ","json":"https://pith.science/pith/X2ORBG53MHPMCDWOGAROX4TEBQ.json","graph_json":"https://pith.science/api/pith-number/X2ORBG53MHPMCDWOGAROX4TEBQ/graph.json","events_json":"https://pith.science/api/pith-number/X2ORBG53MHPMCDWOGAROX4TEBQ/events.json","paper":"https://pith.science/paper/X2ORBG53"},"agent_actions":{"view_html":"https://pith.science/pith/X2ORBG53MHPMCDWOGAROX4TEBQ","download_json":"https://pith.science/pith/X2ORBG53MHPMCDWOGAROX4TEBQ.json","view_paper":"https://pith.science/paper/X2ORBG53","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.03574&json=true","fetch_graph":"https://pith.science/api/pith-number/X2ORBG53MHPMCDWOGAROX4TEBQ/graph.json","fetch_events":"https://pith.science/api/pith-number/X2ORBG53MHPMCDWOGAROX4TEBQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/X2ORBG53MHPMCDWOGAROX4TEBQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/X2ORBG53MHPMCDWOGAROX4TEBQ/action/storage_attestation","attest_author":"https://pith.science/pith/X2ORBG53MHPMCDWOGAROX4TEBQ/action/author_attestation","sign_citation":"https://pith.science/pith/X2ORBG53MHPMCDWOGAROX4TEBQ/action/citation_signature","submit_replication":"https://pith.science/pith/X2ORBG53MHPMCDWOGAROX4TEBQ/action/replication_record"}},"created_at":"2026-08-05T01:36:40.490446+00:00","updated_at":"2026-08-05T01:36:40.490446+00:00"}