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We show that (1) if $\\|A\\|=1$ and $w(A\\otimes B)=w(B)$, then either $A$ has a unitary part or $A$ is completely nonunitary and the numerical range $W(B)$ of $B$ is a circular disc centered at the origin, (2) if $\\|A\\|=\\|A^k\\|=1$ for some $k$, $1\\le k<\\infty$, then $w(A)\\ge\\cos(\\pi/(k+2))$, and, moreover"},"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":"1306.2423","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-06-11T05:31:18Z","cross_cats_sorted":[],"title_canon_sha256":"76237cac9c48ee7910cfe5177bbcf9726f2556d957eb888ce3f5c0305652b063","abstract_canon_sha256":"5d9be0acf81b3ab0be83911b7e58b18a146a7e89f47ffcce7a60df6eb6b82544"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:48.278375Z","signature_b64":"syQHGOiFcWo1zpVrprrRFuV0KBqsQqH8lplzh725ObTq76vcTh65JpuhVdDpSw2eTOgOSfqnTd5TI2T5xDiADg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85498163b7e2f507b10012c7cc44424ec8e115864d82907811eb98ef4e19607c","last_reissued_at":"2026-05-18T03:09:48.277477Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:48.277477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Numerical Radii for Tensor Products of Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hwa-Long Gau, Kuo-Zhong Wang, Pei Yuan Wu","submitted_at":"2013-06-11T05:31:18Z","abstract_excerpt":"For $n$-by-$n$ and $m$-by-$m$ complex matrices $A$ and $B$, it is known that the inequality $w(A\\otimes B)\\le\\|A\\|w(B)$ holds, where $w(\\cdot)$ and $\\|\\cdot\\|$ denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if $\\|A\\|=1$ and $w(A\\otimes B)=w(B)$, then either $A$ has a unitary part or $A$ is completely nonunitary and the numerical range $W(B)$ of $B$ is a circular disc centered at the origin, (2) if $\\|A\\|=\\|A^k\\|=1$ for some $k$, $1\\le k<\\infty$, then $w(A)\\ge\\cos(\\pi/(k+2))$, and, moreover"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.2423","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":"1306.2423","created_at":"2026-05-18T03:09:48.277641+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.2423v1","created_at":"2026-05-18T03:09:48.277641+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.2423","created_at":"2026-05-18T03:09:48.277641+00:00"},{"alias_kind":"pith_short_12","alias_value":"QVEYCY5X4L2Q","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_16","alias_value":"QVEYCY5X4L2QPMIA","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_8","alias_value":"QVEYCY5X","created_at":"2026-05-18T12:27:57.521954+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/QVEYCY5X4L2QPMIACLD4YRCCJ3","json":"https://pith.science/pith/QVEYCY5X4L2QPMIACLD4YRCCJ3.json","graph_json":"https://pith.science/api/pith-number/QVEYCY5X4L2QPMIACLD4YRCCJ3/graph.json","events_json":"https://pith.science/api/pith-number/QVEYCY5X4L2QPMIACLD4YRCCJ3/events.json","paper":"https://pith.science/paper/QVEYCY5X"},"agent_actions":{"view_html":"https://pith.science/pith/QVEYCY5X4L2QPMIACLD4YRCCJ3","download_json":"https://pith.science/pith/QVEYCY5X4L2QPMIACLD4YRCCJ3.json","view_paper":"https://pith.science/paper/QVEYCY5X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.2423&json=true","fetch_graph":"https://pith.science/api/pith-number/QVEYCY5X4L2QPMIACLD4YRCCJ3/graph.json","fetch_events":"https://pith.science/api/pith-number/QVEYCY5X4L2QPMIACLD4YRCCJ3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QVEYCY5X4L2QPMIACLD4YRCCJ3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QVEYCY5X4L2QPMIACLD4YRCCJ3/action/storage_attestation","attest_author":"https://pith.science/pith/QVEYCY5X4L2QPMIACLD4YRCCJ3/action/author_attestation","sign_citation":"https://pith.science/pith/QVEYCY5X4L2QPMIACLD4YRCCJ3/action/citation_signature","submit_replication":"https://pith.science/pith/QVEYCY5X4L2QPMIACLD4YRCCJ3/action/replication_record"}},"created_at":"2026-05-18T03:09:48.277641+00:00","updated_at":"2026-05-18T03:09:48.277641+00:00"}