{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:Q75MPN4YP34APIZQB7XIAOUJUA","short_pith_number":"pith:Q75MPN4Y","canonical_record":{"source":{"id":"1406.7546","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-29T19:42:58Z","cross_cats_sorted":[],"title_canon_sha256":"dd7517649a5855110a7678b8da7e48c39496dea0c80b43cdac8bf5cbab3a8f03","abstract_canon_sha256":"cb98fd8c134c84136df28dc2626516638329f44ef22ac6cbaf201dc957b248dd"},"schema_version":"1.0"},"canonical_sha256":"87fac7b7987ef807a3300fee803a89a02b22cc47f6304c495c6bcaec5008720f","source":{"kind":"arxiv","id":"1406.7546","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.7546","created_at":"2026-05-18T02:48:44Z"},{"alias_kind":"arxiv_version","alias_value":"1406.7546v1","created_at":"2026-05-18T02:48:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.7546","created_at":"2026-05-18T02:48:44Z"},{"alias_kind":"pith_short_12","alias_value":"Q75MPN4YP34A","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"Q75MPN4YP34APIZQ","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"Q75MPN4Y","created_at":"2026-05-18T12:28:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:Q75MPN4YP34APIZQB7XIAOUJUA","target":"record","payload":{"canonical_record":{"source":{"id":"1406.7546","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-29T19:42:58Z","cross_cats_sorted":[],"title_canon_sha256":"dd7517649a5855110a7678b8da7e48c39496dea0c80b43cdac8bf5cbab3a8f03","abstract_canon_sha256":"cb98fd8c134c84136df28dc2626516638329f44ef22ac6cbaf201dc957b248dd"},"schema_version":"1.0"},"canonical_sha256":"87fac7b7987ef807a3300fee803a89a02b22cc47f6304c495c6bcaec5008720f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:44.918566Z","signature_b64":"glSw0nSZiZj3NB//nN29KJ0RlAbCrMABOeQQtjFuqnnyrSOCCXnAnG3lx3+6zR+V6JMsAJOlfV+SAMFAtWUpDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"87fac7b7987ef807a3300fee803a89a02b22cc47f6304c495c6bcaec5008720f","last_reissued_at":"2026-05-18T02:48:44.917849Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:44.917849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1406.7546","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:48:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g7W11Vbweu9IfvnGeqlvoqrO8gyVght0Dap6TLxokc2Ib2dFUyEsOgLEeMpvZHHPy++7oHyVVzlKlRz7HKbgBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T04:35:23.837746Z"},"content_sha256":"9ed6dd6d4bf65a3aca2214f38b7b8d3d50075c4140e1ea75319d92e7e95a08ac","schema_version":"1.0","event_id":"sha256:9ed6dd6d4bf65a3aca2214f38b7b8d3d50075c4140e1ea75319d92e7e95a08ac"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:Q75MPN4YP34APIZQB7XIAOUJUA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sur quelques extensions au cadre Banachique de la notion d'op\\'erateur de Hilbert-Schmidt","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Bernhard Hermann Haak (IMB), Jean Esterle (IMB), Said Amana Abdillah","submitted_at":"2014-06-29T19:42:58Z","abstract_excerpt":"In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operators: $p$-summing operators, $\\gamma$-summing or $\\gamma$-radonifying operators, weakly $*1$-nuclear operators and classes of operators defined via factorization properties. We introduce the class $PS_2(E; F)$ of pre-Hilbert-Schmidt operators as the class of all operators $u:E\\to F$ such that $w\\circ u \\circ v$ is Hilbert-Schmidt for every bounded operator $v: H_1\\to E$ and every bounded operator $w:F\\to H_2$, where $H_1$ et $H_2$ are Hilbert spaces. Besides the trivial case where "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.7546","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:48:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZfOj+ALelcJH03HCES0JpAsRT4HwoMdSSxr6rpaw5U2ZMNK1VG2LMVfnoP9OxGB0NL5ebn4B7mtZ8rFIrYZ6CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T04:35:23.838091Z"},"content_sha256":"df0604b201735eccb38bedcbba302908339e4203eb2da25145fd9246ba6328d5","schema_version":"1.0","event_id":"sha256:df0604b201735eccb38bedcbba302908339e4203eb2da25145fd9246ba6328d5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q75MPN4YP34APIZQB7XIAOUJUA/bundle.json","state_url":"https://pith.science/pith/Q75MPN4YP34APIZQB7XIAOUJUA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q75MPN4YP34APIZQB7XIAOUJUA/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":"2026-06-02T04:35:23Z","links":{"resolver":"https://pith.science/pith/Q75MPN4YP34APIZQB7XIAOUJUA","bundle":"https://pith.science/pith/Q75MPN4YP34APIZQB7XIAOUJUA/bundle.json","state":"https://pith.science/pith/Q75MPN4YP34APIZQB7XIAOUJUA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q75MPN4YP34APIZQB7XIAOUJUA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:Q75MPN4YP34APIZQB7XIAOUJUA","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":"cb98fd8c134c84136df28dc2626516638329f44ef22ac6cbaf201dc957b248dd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-29T19:42:58Z","title_canon_sha256":"dd7517649a5855110a7678b8da7e48c39496dea0c80b43cdac8bf5cbab3a8f03"},"schema_version":"1.0","source":{"id":"1406.7546","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.7546","created_at":"2026-05-18T02:48:44Z"},{"alias_kind":"arxiv_version","alias_value":"1406.7546v1","created_at":"2026-05-18T02:48:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.7546","created_at":"2026-05-18T02:48:44Z"},{"alias_kind":"pith_short_12","alias_value":"Q75MPN4YP34A","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"Q75MPN4YP34APIZQ","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"Q75MPN4Y","created_at":"2026-05-18T12:28:43Z"}],"graph_snapshots":[{"event_id":"sha256:df0604b201735eccb38bedcbba302908339e4203eb2da25145fd9246ba6328d5","target":"graph","created_at":"2026-05-18T02:48:44Z","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":"In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operators: $p$-summing operators, $\\gamma$-summing or $\\gamma$-radonifying operators, weakly $*1$-nuclear operators and classes of operators defined via factorization properties. We introduce the class $PS_2(E; F)$ of pre-Hilbert-Schmidt operators as the class of all operators $u:E\\to F$ such that $w\\circ u \\circ v$ is Hilbert-Schmidt for every bounded operator $v: H_1\\to E$ and every bounded operator $w:F\\to H_2$, where $H_1$ et $H_2$ are Hilbert spaces. Besides the trivial case where ","authors_text":"Bernhard Hermann Haak (IMB), Jean Esterle (IMB), Said Amana Abdillah","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-29T19:42:58Z","title":"Sur quelques extensions au cadre Banachique de la notion d'op\\'erateur de Hilbert-Schmidt"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.7546","kind":"arxiv","version":1},"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":"sha256:9ed6dd6d4bf65a3aca2214f38b7b8d3d50075c4140e1ea75319d92e7e95a08ac","target":"record","created_at":"2026-05-18T02:48:44Z","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":"cb98fd8c134c84136df28dc2626516638329f44ef22ac6cbaf201dc957b248dd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-06-29T19:42:58Z","title_canon_sha256":"dd7517649a5855110a7678b8da7e48c39496dea0c80b43cdac8bf5cbab3a8f03"},"schema_version":"1.0","source":{"id":"1406.7546","kind":"arxiv","version":1}},"canonical_sha256":"87fac7b7987ef807a3300fee803a89a02b22cc47f6304c495c6bcaec5008720f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"87fac7b7987ef807a3300fee803a89a02b22cc47f6304c495c6bcaec5008720f","first_computed_at":"2026-05-18T02:48:44.917849Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:48:44.917849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"glSw0nSZiZj3NB//nN29KJ0RlAbCrMABOeQQtjFuqnnyrSOCCXnAnG3lx3+6zR+V6JMsAJOlfV+SAMFAtWUpDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:48:44.918566Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.7546","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9ed6dd6d4bf65a3aca2214f38b7b8d3d50075c4140e1ea75319d92e7e95a08ac","sha256:df0604b201735eccb38bedcbba302908339e4203eb2da25145fd9246ba6328d5"],"state_sha256":"34fc9b9f54c4919fd4a59f082aaf447d2282590ef8e79e3d0cffc729df9b9873"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sxnjoQ2Sk6XHm9Ec5z4GrXWStbbsk1Q8mcIE8evavoj6fjI4dmDeArThAbP0D7HPQrWEoG+hl1CLz3TtioQOAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T04:35:23.840166Z","bundle_sha256":"16b0dffe1b183ec8ca0c9246dd1d19bdaeaa463691d46f0cb8d28dd29e40fc19"}}