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Using this model, we show that if $A$ has ascent $k$ and $A, A^2, \\ldots, A^{k-1}$ are partial isometries, then the numerical range $W(A)$ of $A$ is a circular disc centered at the origin if and only if $A$ is unitarily similar to a direct"},"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":"1310.4952","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-10-18T08:54:43Z","cross_cats_sorted":[],"title_canon_sha256":"cd5e58bda9ff4edec32aa0e901a370af6dbc215a249c748b9b583f2df8444152","abstract_canon_sha256":"efff7159a4c3eaf241b48f8d6808c4c239e55c416c4c25bd9b32753fa32af049"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:01.786802Z","signature_b64":"tQUlITjxNeNIbHWpwJOPpO1r1frGSxki7WNBLjqg4qFfneIzV2HU5fudYdU9B2RCLRc/h1oYMxi8DeE21d5mBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"639791be7601585587a244b8ca997b44512b87b9a18a4532b5dfd86eaff99caf","last_reissued_at":"2026-05-18T03:10:01.786227Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:01.786227Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Structures and Numerical Ranges of Power Partial Isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hwa-Long Gau, Pei Yuan Wu","submitted_at":"2013-10-18T08:54:43Z","abstract_excerpt":"We derive a matrix model, under unitary similarity, of an $n$-by-$n$ matrix $A$ such that $A, A^2, \\ldots, A^k$ ($k\\ge 1$) are all partial isometries, which generalizes the known fact that if $A$ is a partial isometry, then it is unitarily similar to a matrix of the form ${\\scriptsize\\left[\\begin{array}{cc} 0 & B 0 & C\\end{array}\\right]}$ with $B^*B+C^*C=I$. Using this model, we show that if $A$ has ascent $k$ and $A, A^2, \\ldots, A^{k-1}$ are partial isometries, then the numerical range $W(A)$ of $A$ is a circular disc centered at the origin if and only if $A$ is unitarily similar to a direct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4952","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":""},"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":"1310.4952","created_at":"2026-05-18T03:10:01.786300+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.4952v1","created_at":"2026-05-18T03:10:01.786300+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.4952","created_at":"2026-05-18T03:10:01.786300+00:00"},{"alias_kind":"pith_short_12","alias_value":"MOLZDPTWAFMF","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_16","alias_value":"MOLZDPTWAFMFLB5C","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_8","alias_value":"MOLZDPTW","created_at":"2026-05-18T12:27:52.871228+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR","json":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR.json","graph_json":"https://pith.science/api/pith-number/MOLZDPTWAFMFLB5CIS4MVGL3IR/graph.json","events_json":"https://pith.science/api/pith-number/MOLZDPTWAFMFLB5CIS4MVGL3IR/events.json","paper":"https://pith.science/paper/MOLZDPTW"},"agent_actions":{"view_html":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR","download_json":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR.json","view_paper":"https://pith.science/paper/MOLZDPTW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.4952&json=true","fetch_graph":"https://pith.science/api/pith-number/MOLZDPTWAFMFLB5CIS4MVGL3IR/graph.json","fetch_events":"https://pith.science/api/pith-number/MOLZDPTWAFMFLB5CIS4MVGL3IR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/action/storage_attestation","attest_author":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/action/author_attestation","sign_citation":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/action/citation_signature","submit_replication":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/action/replication_record"}},"created_at":"2026-05-18T03:10:01.786300+00:00","updated_at":"2026-05-18T03:10:01.786300+00:00"}