{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:FGXDRGMGHOVGBQF2CB3IJPRKU3","short_pith_number":"pith:FGXDRGMG","canonical_record":{"source":{"id":"0901.1771","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-01-13T12:15:30Z","cross_cats_sorted":[],"title_canon_sha256":"7a9a2d464f3c9392fd7f2d865516543020e4c8024202072967415f27ff4daead","abstract_canon_sha256":"fdb87ac029a45327fce947d2a92ee17d4ce749ece0f9d7c2e3a2fba720b997e2"},"schema_version":"1.0"},"canonical_sha256":"29ae3899863baa60c0ba107684be2aa6dd86870ff040b6173b523946a0b7b145","source":{"kind":"arxiv","id":"0901.1771","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.1771","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"arxiv_version","alias_value":"0901.1771v2","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.1771","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"pith_short_12","alias_value":"FGXDRGMGHOVG","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"FGXDRGMGHOVGBQF2","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"FGXDRGMG","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:FGXDRGMGHOVGBQF2CB3IJPRKU3","target":"record","payload":{"canonical_record":{"source":{"id":"0901.1771","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-01-13T12:15:30Z","cross_cats_sorted":[],"title_canon_sha256":"7a9a2d464f3c9392fd7f2d865516543020e4c8024202072967415f27ff4daead","abstract_canon_sha256":"fdb87ac029a45327fce947d2a92ee17d4ce749ece0f9d7c2e3a2fba720b997e2"},"schema_version":"1.0"},"canonical_sha256":"29ae3899863baa60c0ba107684be2aa6dd86870ff040b6173b523946a0b7b145","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:39.901006Z","signature_b64":"9vy7YNaZH/pORBEbY6JdRASSbYSwU0lXjzozNiDnIDyHkG6kH/N3el+GPKc2gC7baXAvTkzqUazlrBpOh3SfBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29ae3899863baa60c0ba107684be2aa6dd86870ff040b6173b523946a0b7b145","last_reissued_at":"2026-05-18T02:54:39.900532Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:39.900532Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0901.1771","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:54:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zGTEtvCV4NZzrac+eWPqThKGhG+koz+NHVLo/m3u0cUlNSyb/06/u+OxGWLcLd9zOHT2UViKnRpwXlWEMg/3BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:06:15.065815Z"},"content_sha256":"a14c2dcddf199c37ecbb8329660a405b2bc2c503fb3202f65b79213f199c354a","schema_version":"1.0","event_id":"sha256:a14c2dcddf199c37ecbb8329660a405b2bc2c503fb3202f65b79213f199c354a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:FGXDRGMGHOVGBQF2CB3IJPRKU3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Pettis-Type Integral and Applications to Transition Semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Markus Kunze","submitted_at":"2009-01-13T12:15:30Z","abstract_excerpt":"Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we give conditions ensuring that a semigroup consisting of kernel operators has a Laplace transform which also consists of kernel operators. We also provide conditions under which a semigroup is uniquely determined by its Laplace transform"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.1771","kind":"arxiv","version":2},"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":"2026-05-18T02:54:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t2JPaR1OgmMMPRNlvfxxWYLO7vMx3D/uAa1qNmgATN42KLLicz3QKCLtNWvIwFRbQvQBMz9fJaPxIWIHonMWCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:06:15.066473Z"},"content_sha256":"2e0ca2fcf5a2f6d0376d20aea0b2b6d994a177f43c8953c8e1a7ea6641841879","schema_version":"1.0","event_id":"sha256:2e0ca2fcf5a2f6d0376d20aea0b2b6d994a177f43c8953c8e1a7ea6641841879"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FGXDRGMGHOVGBQF2CB3IJPRKU3/bundle.json","state_url":"https://pith.science/pith/FGXDRGMGHOVGBQF2CB3IJPRKU3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FGXDRGMGHOVGBQF2CB3IJPRKU3/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":"2026-05-27T00:06:15Z","links":{"resolver":"https://pith.science/pith/FGXDRGMGHOVGBQF2CB3IJPRKU3","bundle":"https://pith.science/pith/FGXDRGMGHOVGBQF2CB3IJPRKU3/bundle.json","state":"https://pith.science/pith/FGXDRGMGHOVGBQF2CB3IJPRKU3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FGXDRGMGHOVGBQF2CB3IJPRKU3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:FGXDRGMGHOVGBQF2CB3IJPRKU3","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":"fdb87ac029a45327fce947d2a92ee17d4ce749ece0f9d7c2e3a2fba720b997e2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-01-13T12:15:30Z","title_canon_sha256":"7a9a2d464f3c9392fd7f2d865516543020e4c8024202072967415f27ff4daead"},"schema_version":"1.0","source":{"id":"0901.1771","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.1771","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"arxiv_version","alias_value":"0901.1771v2","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.1771","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"pith_short_12","alias_value":"FGXDRGMGHOVG","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"FGXDRGMGHOVGBQF2","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"FGXDRGMG","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:2e0ca2fcf5a2f6d0376d20aea0b2b6d994a177f43c8953c8e1a7ea6641841879","target":"graph","created_at":"2026-05-18T02:54:39Z","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":"Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we give conditions ensuring that a semigroup consisting of kernel operators has a Laplace transform which also consists of kernel operators. We also provide conditions under which a semigroup is uniquely determined by its Laplace transform","authors_text":"Markus Kunze","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-01-13T12:15:30Z","title":"A Pettis-Type Integral and Applications to Transition Semigroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.1771","kind":"arxiv","version":2},"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:a14c2dcddf199c37ecbb8329660a405b2bc2c503fb3202f65b79213f199c354a","target":"record","created_at":"2026-05-18T02:54:39Z","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":"fdb87ac029a45327fce947d2a92ee17d4ce749ece0f9d7c2e3a2fba720b997e2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-01-13T12:15:30Z","title_canon_sha256":"7a9a2d464f3c9392fd7f2d865516543020e4c8024202072967415f27ff4daead"},"schema_version":"1.0","source":{"id":"0901.1771","kind":"arxiv","version":2}},"canonical_sha256":"29ae3899863baa60c0ba107684be2aa6dd86870ff040b6173b523946a0b7b145","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29ae3899863baa60c0ba107684be2aa6dd86870ff040b6173b523946a0b7b145","first_computed_at":"2026-05-18T02:54:39.900532Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:54:39.900532Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9vy7YNaZH/pORBEbY6JdRASSbYSwU0lXjzozNiDnIDyHkG6kH/N3el+GPKc2gC7baXAvTkzqUazlrBpOh3SfBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:54:39.901006Z","signed_message":"canonical_sha256_bytes"},"source_id":"0901.1771","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a14c2dcddf199c37ecbb8329660a405b2bc2c503fb3202f65b79213f199c354a","sha256:2e0ca2fcf5a2f6d0376d20aea0b2b6d994a177f43c8953c8e1a7ea6641841879"],"state_sha256":"ba142643ed4dba64b91923ce03f1bae78554bf92446129af80d590534e43d2b8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XU8Rt/X4rfL91+OOECBd5pec6kCyQ+grFPw1M31DabcNm2arX3wvwnqm3HNwuhIjqKfDs4McE3obMAZBlyiTDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T00:06:15.069765Z","bundle_sha256":"6583e5d3c8e575904a31b7efe18d08baa35b284ab28132c286d82fb8a3b92230"}}