{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:23RTLQ4KLA3BQLRKQXADYS7YS4","short_pith_number":"pith:23RTLQ4K","canonical_record":{"source":{"id":"1211.5753","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-25T10:14:53Z","cross_cats_sorted":[],"title_canon_sha256":"942659bdd51525a9daa4d1afd6bf1d85dba57a7d457f76a8319b40285a5f2b62","abstract_canon_sha256":"646c64db8498fa2f4f9ff0be02dc65510deb7860139ea5b5a3c0da33269b8829"},"schema_version":"1.0"},"canonical_sha256":"d6e335c38a5836182e2a85c03c4bf8970fe5d7682612d6a15c07659b12d70a5c","source":{"kind":"arxiv","id":"1211.5753","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.5753","created_at":"2026-05-18T03:40:02Z"},{"alias_kind":"arxiv_version","alias_value":"1211.5753v1","created_at":"2026-05-18T03:40:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5753","created_at":"2026-05-18T03:40:02Z"},{"alias_kind":"pith_short_12","alias_value":"23RTLQ4KLA3B","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"23RTLQ4KLA3BQLRK","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"23RTLQ4K","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:23RTLQ4KLA3BQLRKQXADYS7YS4","target":"record","payload":{"canonical_record":{"source":{"id":"1211.5753","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-25T10:14:53Z","cross_cats_sorted":[],"title_canon_sha256":"942659bdd51525a9daa4d1afd6bf1d85dba57a7d457f76a8319b40285a5f2b62","abstract_canon_sha256":"646c64db8498fa2f4f9ff0be02dc65510deb7860139ea5b5a3c0da33269b8829"},"schema_version":"1.0"},"canonical_sha256":"d6e335c38a5836182e2a85c03c4bf8970fe5d7682612d6a15c07659b12d70a5c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:02.031839Z","signature_b64":"wYHpGi3DbJkWbNXHQJOasQvep1Hc5UdKtwLZvL1z5S5Mk1UEtP9p8380bis7blW5mvMGZkbQjH+EPy9kwjWuCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6e335c38a5836182e2a85c03c4bf8970fe5d7682612d6a15c07659b12d70a5c","last_reissued_at":"2026-05-18T03:40:02.031248Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:02.031248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.5753","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":"2026-05-18T03:40:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GjxHdFjd79aIRzJrttvR7dwrVDYTxEezwVoLJ1NdmE+ocp86M7z+j7VlerxRnvjm8HBT/D8A6LZlqOZ1HO8hBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:16:40.258935Z"},"content_sha256":"0d4139141cee7f97450a5b31d16443dce97dc6415248bf290d37cfc06d9f1ab4","schema_version":"1.0","event_id":"sha256:0d4139141cee7f97450a5b31d16443dce97dc6415248bf290d37cfc06d9f1ab4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:23RTLQ4KLA3BQLRKQXADYS7YS4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the numerical radius of Lipschitz operators in Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Dongni Tan, Ruidong Wang, Xujian Huang","submitted_at":"2012-11-25T10:14:53Z","abstract_excerpt":"We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and $C$-rich subspaces have Lipschitz numerical index 1. Moreover, using the G$\\hat{a}$teaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Fi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5753","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":"2026-05-18T03:40:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u9DwW2ITbPlwet+NyBf5BRtLR/Ta8Rtu4hi/qzY23JVyHuZhVuAVgrUoygDv0SDarN6sh+jQ3+53XfDwujuyBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:16:40.259609Z"},"content_sha256":"7c9c74adb6e5140bff0d5e014a8f6b48703083b6ae1f5e2f2c70000b86b2d2b6","schema_version":"1.0","event_id":"sha256:7c9c74adb6e5140bff0d5e014a8f6b48703083b6ae1f5e2f2c70000b86b2d2b6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/23RTLQ4KLA3BQLRKQXADYS7YS4/bundle.json","state_url":"https://pith.science/pith/23RTLQ4KLA3BQLRKQXADYS7YS4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/23RTLQ4KLA3BQLRKQXADYS7YS4/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":"2026-05-31T18:16:40Z","links":{"resolver":"https://pith.science/pith/23RTLQ4KLA3BQLRKQXADYS7YS4","bundle":"https://pith.science/pith/23RTLQ4KLA3BQLRKQXADYS7YS4/bundle.json","state":"https://pith.science/pith/23RTLQ4KLA3BQLRKQXADYS7YS4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/23RTLQ4KLA3BQLRKQXADYS7YS4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:23RTLQ4KLA3BQLRKQXADYS7YS4","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":"646c64db8498fa2f4f9ff0be02dc65510deb7860139ea5b5a3c0da33269b8829","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-25T10:14:53Z","title_canon_sha256":"942659bdd51525a9daa4d1afd6bf1d85dba57a7d457f76a8319b40285a5f2b62"},"schema_version":"1.0","source":{"id":"1211.5753","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.5753","created_at":"2026-05-18T03:40:02Z"},{"alias_kind":"arxiv_version","alias_value":"1211.5753v1","created_at":"2026-05-18T03:40:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5753","created_at":"2026-05-18T03:40:02Z"},{"alias_kind":"pith_short_12","alias_value":"23RTLQ4KLA3B","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"23RTLQ4KLA3BQLRK","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"23RTLQ4K","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:7c9c74adb6e5140bff0d5e014a8f6b48703083b6ae1f5e2f2c70000b86b2d2b6","target":"graph","created_at":"2026-05-18T03:40:02Z","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":"We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and $C$-rich subspaces have Lipschitz numerical index 1. Moreover, using the G$\\hat{a}$teaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Fi","authors_text":"Dongni Tan, Ruidong Wang, Xujian Huang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-25T10:14:53Z","title":"On the numerical radius of Lipschitz operators in Banach spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5753","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":"sha256:0d4139141cee7f97450a5b31d16443dce97dc6415248bf290d37cfc06d9f1ab4","target":"record","created_at":"2026-05-18T03:40:02Z","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":"646c64db8498fa2f4f9ff0be02dc65510deb7860139ea5b5a3c0da33269b8829","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-25T10:14:53Z","title_canon_sha256":"942659bdd51525a9daa4d1afd6bf1d85dba57a7d457f76a8319b40285a5f2b62"},"schema_version":"1.0","source":{"id":"1211.5753","kind":"arxiv","version":1}},"canonical_sha256":"d6e335c38a5836182e2a85c03c4bf8970fe5d7682612d6a15c07659b12d70a5c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6e335c38a5836182e2a85c03c4bf8970fe5d7682612d6a15c07659b12d70a5c","first_computed_at":"2026-05-18T03:40:02.031248Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:40:02.031248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wYHpGi3DbJkWbNXHQJOasQvep1Hc5UdKtwLZvL1z5S5Mk1UEtP9p8380bis7blW5mvMGZkbQjH+EPy9kwjWuCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:40:02.031839Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.5753","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0d4139141cee7f97450a5b31d16443dce97dc6415248bf290d37cfc06d9f1ab4","sha256:7c9c74adb6e5140bff0d5e014a8f6b48703083b6ae1f5e2f2c70000b86b2d2b6"],"state_sha256":"b786bb3df1b23a39fd3eba396d665079946edaf4d48b836a821cc4668aff82d5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Q+qephZwyFhKGDUUdLRvlME8RsIEAVM8w2gLnxf7pmvHgBgdilta5tZgUssYLD0lseYOWP7//TfgpRh0X/rDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T18:16:40.263227Z","bundle_sha256":"11ac389d69a9d74e5df1edce41573473fe6bd185c1fd2b0f8acba5be941a54bc"}}