{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:PT26T2DHDANJCIPP2TT7EEWORC","short_pith_number":"pith:PT26T2DH","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"},"canonical_sha256":"7cf5e9e867181a9121efd4e7f212ce88853284452a1ca13a68fb07273bf86ac3","source":{"kind":"arxiv","id":"1305.4460","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.4460","created_at":"2026-05-18T03:07:01Z"},{"alias_kind":"arxiv_version","alias_value":"1305.4460v6","created_at":"2026-05-18T03:07:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.4460","created_at":"2026-05-18T03:07:01Z"},{"alias_kind":"pith_short_12","alias_value":"PT26T2DHDANJ","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"PT26T2DHDANJCIPP","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"PT26T2DH","created_at":"2026-05-18T12:27:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:PT26T2DHDANJCIPP2TT7EEWORC","target":"record","payload":{"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"},"canonical_sha256":"7cf5e9e867181a9121efd4e7f212ce88853284452a1ca13a68fb07273bf86ac3","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"},"source_kind":"arxiv","source_id":"1305.4460","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nlxNidRYW5FJQ5+IbvfeskErOzm9k9W2b2ZjlzVYjZMy7TYZq+i+o9mhE1KVLOth+ME+CLOutO5m7iHqHjCHCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T14:04:09.646738Z"},"content_sha256":"db10aa5a585bf56dda288957d926c43f7912e8db4ff6d472ebb2421adf904535","schema_version":"1.0","event_id":"sha256:db10aa5a585bf56dda288957d926c43f7912e8db4ff6d472ebb2421adf904535"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:PT26T2DHDANJCIPP2TT7EEWORC","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HAVqMcQb9JP7QFM/omHuyuZHcCyVysTo6ZEYWOgKrrzyjaJemf5eDUUN6aL9zZFFRqu1PVBOe6MhiQiZg18gDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T14:04:09.647479Z"},"content_sha256":"405bbea0cb7561ac462f6d7b7d60b28eb36f45b9671d3fd9ac630c2fa056beb6","schema_version":"1.0","event_id":"sha256:405bbea0cb7561ac462f6d7b7d60b28eb36f45b9671d3fd9ac630c2fa056beb6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC/bundle.json","state_url":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PT26T2DHDANJCIPP2TT7EEWORC/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-23T14:04:09Z","links":{"resolver":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC","bundle":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC/bundle.json","state":"https://pith.science/pith/PT26T2DHDANJCIPP2TT7EEWORC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PT26T2DHDANJCIPP2TT7EEWORC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:PT26T2DHDANJCIPP2TT7EEWORC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a6a10b56661007af428fd9db83fc23212ce8e66cb1a6c3cd8529a4d6cc68b01f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-05-20T08:10:53Z","title_canon_sha256":"97ed25b499e58887d8444a3f0459820f449cf48ed073242669ed0541f19dbbf3"},"schema_version":"1.0","source":{"id":"1305.4460","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.4460","created_at":"2026-05-18T03:07:01Z"},{"alias_kind":"arxiv_version","alias_value":"1305.4460v6","created_at":"2026-05-18T03:07:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.4460","created_at":"2026-05-18T03:07:01Z"},{"alias_kind":"pith_short_12","alias_value":"PT26T2DHDANJ","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"PT26T2DHDANJCIPP","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"PT26T2DH","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:405bbea0cb7561ac462f6d7b7d60b28eb36f45b9671d3fd9ac630c2fa056beb6","target":"graph","created_at":"2026-05-18T03:07:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Feng-Yu Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-05-20T08:10:53Z","title":"Criteria of Spectral Gap for Markov Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4460","kind":"arxiv","version":6},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:db10aa5a585bf56dda288957d926c43f7912e8db4ff6d472ebb2421adf904535","target":"record","created_at":"2026-05-18T03:07:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a6a10b56661007af428fd9db83fc23212ce8e66cb1a6c3cd8529a4d6cc68b01f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-05-20T08:10:53Z","title_canon_sha256":"97ed25b499e58887d8444a3f0459820f449cf48ed073242669ed0541f19dbbf3"},"schema_version":"1.0","source":{"id":"1305.4460","kind":"arxiv","version":6}},"canonical_sha256":"7cf5e9e867181a9121efd4e7f212ce88853284452a1ca13a68fb07273bf86ac3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7cf5e9e867181a9121efd4e7f212ce88853284452a1ca13a68fb07273bf86ac3","first_computed_at":"2026-05-18T03:07:01.786543Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:01.786543Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"46GH126tBDUmbi1dAo3Z75EBHFYBK7rvUiNGLmBl/NUkHa/9vxIel46kQ36FDn4Zeu3pEL3/6I0hHPH4fo/oAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:01.787181Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.4460","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:db10aa5a585bf56dda288957d926c43f7912e8db4ff6d472ebb2421adf904535","sha256:405bbea0cb7561ac462f6d7b7d60b28eb36f45b9671d3fd9ac630c2fa056beb6"],"state_sha256":"c3fa51a6a7912ff78e30b3ae96484576576bd6c48429141a89854bcc034a9f37"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vepidqmT0Hxey103DQmBlM+Ack5K+eSu5cn20iizTYI+RVm25BcWuTygkNsVyH5RTCMNX3styDEZYdT8mqpfDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T14:04:09.650882Z","bundle_sha256":"1b8ae841efc34dfbd9e407af3dea1997ced867bb7b99c07b8ac3847e6d29f9fb"}}