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More precisely, we consider operators $T$ and $S$ which satisfy $\\omega(T + \\lambda S) = \\omega(T)+\\omega(S)$ for some complex unit $\\lambda$. We show that $T \\parallel_{\\omega} S$ if and only if there exists a sequence of unit vectors $\\{x_n\\}$ in $\\mathscr{H}$ such that \\begin{align*} \\lim_{n\\rightarrow\\infty} \\big|\\langle Tx_n, x_n\\rangle\\langle Sx_n, x_n\\rangle\\big| = \\omega(T)\\omega(S). \\end{align*} We then a"},"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":"1810.10445","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-10-24T15:17:32Z","cross_cats_sorted":[],"title_canon_sha256":"63a24fd55879589c00cb45c8338fa6735c8e2e1bae34ff10db31b8b64f89b1ce","abstract_canon_sha256":"a5a9d2e4da1571900630b7a664826839237901c5fae7a247979fbcdbc0d3b9cd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:23.085015Z","signature_b64":"L643HgQ2StBusKgP9z+XGmJDwwjfwdIz8Nq7jU5/xRhpn3aPlvHkn/saZOqed77TPmnuLsqFT5xzoHxLhiOVCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d38164d3710fce1ab969da0655a22968c25f29abdcba91414c4d7604c685f554","last_reissued_at":"2026-05-18T00:02:23.084231Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:23.084231Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Numerical radius parallelism of Hilbert space operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ali Zamani, Maryam Amyari, Marzieh Mehrazin","submitted_at":"2018-10-24T15:17:32Z","abstract_excerpt":"In this paper, we introduce a new type of parallelism for bounded linear operators on a Hilbert space $\\big(\\mathscr{H}, \\langle \\cdot ,\\cdot \\rangle\\big)$ based on numerical radius. More precisely, we consider operators $T$ and $S$ which satisfy $\\omega(T + \\lambda S) = \\omega(T)+\\omega(S)$ for some complex unit $\\lambda$. We show that $T \\parallel_{\\omega} S$ if and only if there exists a sequence of unit vectors $\\{x_n\\}$ in $\\mathscr{H}$ such that \\begin{align*} \\lim_{n\\rightarrow\\infty} \\big|\\langle Tx_n, x_n\\rangle\\langle Sx_n, x_n\\rangle\\big| = \\omega(T)\\omega(S). \\end{align*} We then a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.10445","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":"1810.10445","created_at":"2026-05-18T00:02:23.084377+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.10445v1","created_at":"2026-05-18T00:02:23.084377+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.10445","created_at":"2026-05-18T00:02:23.084377+00:00"},{"alias_kind":"pith_short_12","alias_value":"2OAWJU3RB7HB","created_at":"2026-05-18T12:32:02.567920+00:00"},{"alias_kind":"pith_short_16","alias_value":"2OAWJU3RB7HBVOLJ","created_at":"2026-05-18T12:32:02.567920+00:00"},{"alias_kind":"pith_short_8","alias_value":"2OAWJU3R","created_at":"2026-05-18T12:32:02.567920+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/2OAWJU3RB7HBVOLJ3IDFLIRJND","json":"https://pith.science/pith/2OAWJU3RB7HBVOLJ3IDFLIRJND.json","graph_json":"https://pith.science/api/pith-number/2OAWJU3RB7HBVOLJ3IDFLIRJND/graph.json","events_json":"https://pith.science/api/pith-number/2OAWJU3RB7HBVOLJ3IDFLIRJND/events.json","paper":"https://pith.science/paper/2OAWJU3R"},"agent_actions":{"view_html":"https://pith.science/pith/2OAWJU3RB7HBVOLJ3IDFLIRJND","download_json":"https://pith.science/pith/2OAWJU3RB7HBVOLJ3IDFLIRJND.json","view_paper":"https://pith.science/paper/2OAWJU3R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.10445&json=true","fetch_graph":"https://pith.science/api/pith-number/2OAWJU3RB7HBVOLJ3IDFLIRJND/graph.json","fetch_events":"https://pith.science/api/pith-number/2OAWJU3RB7HBVOLJ3IDFLIRJND/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2OAWJU3RB7HBVOLJ3IDFLIRJND/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2OAWJU3RB7HBVOLJ3IDFLIRJND/action/storage_attestation","attest_author":"https://pith.science/pith/2OAWJU3RB7HBVOLJ3IDFLIRJND/action/author_attestation","sign_citation":"https://pith.science/pith/2OAWJU3RB7HBVOLJ3IDFLIRJND/action/citation_signature","submit_replication":"https://pith.science/pith/2OAWJU3RB7HBVOLJ3IDFLIRJND/action/replication_record"}},"created_at":"2026-05-18T00:02:23.084377+00:00","updated_at":"2026-05-18T00:02:23.084377+00:00"}