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Then $\\hat P:=\\ff 1 2 (P+P^*)$ has a spectral gap, i.e. $1$ is isolated in the spectrum of $\\hat P$, if and only if $$\\|P\\|_\\tau:=\\lim_{R\\to\\infty} \\sup_{\\mu(f^2)\\le 1}\\mu\\big(f(Pf-R)^+\\big)<1.$$ This strengthens a conjecture of Simon and H$\\phi$egh-Krohn on the spectral gap for hyperbounded operators solved recently by L. Miclo in \\cite{M}. Consequently, for a symmetric, conservative, irreduc"},"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":"1305.4460","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-05-20T08:10:53Z","cross_cats_sorted":[],"title_canon_sha256":"97ed25b499e58887d8444a3f0459820f449cf48ed073242669ed0541f19dbbf3","abstract_canon_sha256":"a6a10b56661007af428fd9db83fc23212ce8e66cb1a6c3cd8529a4d6cc68b01f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:01.787181Z","signature_b64":"46GH126tBDUmbi1dAo3Z75EBHFYBK7rvUiNGLmBl/NUkHa/9vxIel46kQ36FDn4Zeu3pEL3/6I0hHPH4fo/oAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7cf5e9e867181a9121efd4e7f212ce88853284452a1ca13a68fb07273bf86ac3","last_reissued_at":"2026-05-18T03:07:01.786543Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:01.786543Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Criteria of Spectral Gap for Markov Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Feng-Yu Wang","submitted_at":"2013-05-20T08:10:53Z","abstract_excerpt":"Let $(E,\\mathcal F,\\mu)$ be a probability space, and let $P$ be a Markov operator on $L^2(\\mu)$ with $1$ a simple eigenvalue such that $\\mu P=\\mu$ (i.e. $\\mu$ is an invariant probability measure of $P$). Then $\\hat P:=\\ff 1 2 (P+P^*)$ has a spectral gap, i.e. $1$ is isolated in the spectrum of $\\hat P$, if and only if $$\\|P\\|_\\tau:=\\lim_{R\\to\\infty} \\sup_{\\mu(f^2)\\le 1}\\mu\\big(f(Pf-R)^+\\big)<1.$$ This strengthens a conjecture of Simon and H$\\phi$egh-Krohn on the spectral gap for hyperbounded operators solved recently by L. Miclo in \\cite{M}. Consequently, for a symmetric, conservative, irreduc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4460","kind":"arxiv","version":6},"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":"1305.4460","created_at":"2026-05-18T03:07:01.786636+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.4460v6","created_at":"2026-05-18T03:07:01.786636+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.4460","created_at":"2026-05-18T03:07:01.786636+00:00"},{"alias_kind":"pith_short_12","alias_value":"PT26T2DHDANJ","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_16","alias_value":"PT26T2DHDANJCIPP","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_8","alias_value":"PT26T2DH","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/PT26T2DHDANJCIPP2TT7EEWORC","json":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC.json","graph_json":"https://pith.science/api/pith-number/PT26T2DHDANJCIPP2TT7EEWORC/graph.json","events_json":"https://pith.science/api/pith-number/PT26T2DHDANJCIPP2TT7EEWORC/events.json","paper":"https://pith.science/paper/PT26T2DH"},"agent_actions":{"view_html":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC","download_json":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC.json","view_paper":"https://pith.science/paper/PT26T2DH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.4460&json=true","fetch_graph":"https://pith.science/api/pith-number/PT26T2DHDANJCIPP2TT7EEWORC/graph.json","fetch_events":"https://pith.science/api/pith-number/PT26T2DHDANJCIPP2TT7EEWORC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC/action/storage_attestation","attest_author":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC/action/author_attestation","sign_citation":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC/action/citation_signature","submit_replication":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC/action/replication_record"}},"created_at":"2026-05-18T03:07:01.786636+00:00","updated_at":"2026-05-18T03:07:01.786636+00:00"}