{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:B4WL6EBXFWQFRNIRB7KCS3TNA5","short_pith_number":"pith:B4WL6EBX","canonical_record":{"source":{"id":"1209.5050","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-23T11:13:09Z","cross_cats_sorted":[],"title_canon_sha256":"04613ec94c6bb2e8e67ca8a0c5c94e5693baa8f56834b18a42f132b33978b90f","abstract_canon_sha256":"9dec7245422282f6ab52ae1eac9d476c0a10c412eecd01b987eae80a4e5dacc9"},"schema_version":"1.0"},"canonical_sha256":"0f2cbf10372da058b5110fd4296e6d07713333fd8867cdac1768946f44971f00","source":{"kind":"arxiv","id":"1209.5050","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.5050","created_at":"2026-05-18T02:44:32Z"},{"alias_kind":"arxiv_version","alias_value":"1209.5050v1","created_at":"2026-05-18T02:44:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.5050","created_at":"2026-05-18T02:44:32Z"},{"alias_kind":"pith_short_12","alias_value":"B4WL6EBXFWQF","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"B4WL6EBXFWQFRNIR","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"B4WL6EBX","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:B4WL6EBXFWQFRNIRB7KCS3TNA5","target":"record","payload":{"canonical_record":{"source":{"id":"1209.5050","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-23T11:13:09Z","cross_cats_sorted":[],"title_canon_sha256":"04613ec94c6bb2e8e67ca8a0c5c94e5693baa8f56834b18a42f132b33978b90f","abstract_canon_sha256":"9dec7245422282f6ab52ae1eac9d476c0a10c412eecd01b987eae80a4e5dacc9"},"schema_version":"1.0"},"canonical_sha256":"0f2cbf10372da058b5110fd4296e6d07713333fd8867cdac1768946f44971f00","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:32.546247Z","signature_b64":"CmD/lx6KPdtJzlDEni02cM6a6QqLgUEBCYInunzfUhKd5d5L/hbpwWqh2H0Dt1hlvnlCkBO4qV7pLOF+dSUVDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f2cbf10372da058b5110fd4296e6d07713333fd8867cdac1768946f44971f00","last_reissued_at":"2026-05-18T02:44:32.545834Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:32.545834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.5050","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":"2026-05-18T02:44:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TQrdrKOLu6CIa6g5ydqgyQmCMWKGt8ePcJfyVLXO3woOljDCJI3sppdmIMb6ZFj625Exv9Q/ysJ7aYc5mIADDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T14:50:14.387619Z"},"content_sha256":"f0922ca05a393e128c7daecc7b551c157470cdaf0981421e14bc9969c069bfd9","schema_version":"1.0","event_id":"sha256:f0922ca05a393e128c7daecc7b551c157470cdaf0981421e14bc9969c069bfd9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:B4WL6EBXFWQFRNIRB7KCS3TNA5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On $(n,k)$-quasi-*-paranormal operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Huaijie Zhong, Qingping Zeng","submitted_at":"2012-09-23T11:13:09Z","abstract_excerpt":"For nonnegative integers $n$ and $k$, we introduce in this paper a new class of $(n,k)$-quasi-*-paranormal operators satisfying\n  $$||T^{1+n}(T^{k}x)||^{1/(1+n)}||T^{k}x||^{n/(1+n)} \\geq\n  ||T^*(T^{k}x)|| \\makebox{\\ for all} x \\in H.$$\n  This class includes the class of $n$-*-paranormal operators and the class of $(1,k)$-quasi-*-paranormal operators contains the class of $k$-quasi-*-class $A$ operators. We study basic properties of $(n,k)$-quasi-*-paranormal operators: (1) inclusion relations and examples; (2) a matrix representation; (3) joint (approximate) point spectrum and single valued ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5050","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":"2026-05-18T02:44:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wfK2kCVLQatj7L+CZFvIMrEAmiAPGht4UlZQOSYPLjLSsDXRyQ8yLzJcgWEIh9v2RCSlqdoTSdEnekMACz9kCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T14:50:14.387979Z"},"content_sha256":"f4c577ad82905be6d8f4f541be99d3b32972600d2bbb4c98db4c86868d28153a","schema_version":"1.0","event_id":"sha256:f4c577ad82905be6d8f4f541be99d3b32972600d2bbb4c98db4c86868d28153a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B4WL6EBXFWQFRNIRB7KCS3TNA5/bundle.json","state_url":"https://pith.science/pith/B4WL6EBXFWQFRNIRB7KCS3TNA5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B4WL6EBXFWQFRNIRB7KCS3TNA5/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":"2026-07-06T14:50:14Z","links":{"resolver":"https://pith.science/pith/B4WL6EBXFWQFRNIRB7KCS3TNA5","bundle":"https://pith.science/pith/B4WL6EBXFWQFRNIRB7KCS3TNA5/bundle.json","state":"https://pith.science/pith/B4WL6EBXFWQFRNIRB7KCS3TNA5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B4WL6EBXFWQFRNIRB7KCS3TNA5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:B4WL6EBXFWQFRNIRB7KCS3TNA5","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":"9dec7245422282f6ab52ae1eac9d476c0a10c412eecd01b987eae80a4e5dacc9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-23T11:13:09Z","title_canon_sha256":"04613ec94c6bb2e8e67ca8a0c5c94e5693baa8f56834b18a42f132b33978b90f"},"schema_version":"1.0","source":{"id":"1209.5050","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.5050","created_at":"2026-05-18T02:44:32Z"},{"alias_kind":"arxiv_version","alias_value":"1209.5050v1","created_at":"2026-05-18T02:44:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.5050","created_at":"2026-05-18T02:44:32Z"},{"alias_kind":"pith_short_12","alias_value":"B4WL6EBXFWQF","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"B4WL6EBXFWQFRNIR","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"B4WL6EBX","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:f4c577ad82905be6d8f4f541be99d3b32972600d2bbb4c98db4c86868d28153a","target":"graph","created_at":"2026-05-18T02:44:32Z","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":"For nonnegative integers $n$ and $k$, we introduce in this paper a new class of $(n,k)$-quasi-*-paranormal operators satisfying\n  $$||T^{1+n}(T^{k}x)||^{1/(1+n)}||T^{k}x||^{n/(1+n)} \\geq\n  ||T^*(T^{k}x)|| \\makebox{\\ for all} x \\in H.$$\n  This class includes the class of $n$-*-paranormal operators and the class of $(1,k)$-quasi-*-paranormal operators contains the class of $k$-quasi-*-class $A$ operators. We study basic properties of $(n,k)$-quasi-*-paranormal operators: (1) inclusion relations and examples; (2) a matrix representation; (3) joint (approximate) point spectrum and single valued ex","authors_text":"Huaijie Zhong, Qingping Zeng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-23T11:13:09Z","title":"On $(n,k)$-quasi-*-paranormal operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5050","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:f0922ca05a393e128c7daecc7b551c157470cdaf0981421e14bc9969c069bfd9","target":"record","created_at":"2026-05-18T02:44:32Z","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":"9dec7245422282f6ab52ae1eac9d476c0a10c412eecd01b987eae80a4e5dacc9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-23T11:13:09Z","title_canon_sha256":"04613ec94c6bb2e8e67ca8a0c5c94e5693baa8f56834b18a42f132b33978b90f"},"schema_version":"1.0","source":{"id":"1209.5050","kind":"arxiv","version":1}},"canonical_sha256":"0f2cbf10372da058b5110fd4296e6d07713333fd8867cdac1768946f44971f00","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0f2cbf10372da058b5110fd4296e6d07713333fd8867cdac1768946f44971f00","first_computed_at":"2026-05-18T02:44:32.545834Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:32.545834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CmD/lx6KPdtJzlDEni02cM6a6QqLgUEBCYInunzfUhKd5d5L/hbpwWqh2H0Dt1hlvnlCkBO4qV7pLOF+dSUVDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:32.546247Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.5050","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0922ca05a393e128c7daecc7b551c157470cdaf0981421e14bc9969c069bfd9","sha256:f4c577ad82905be6d8f4f541be99d3b32972600d2bbb4c98db4c86868d28153a"],"state_sha256":"ff011d41253aa2917dd80a4853bcd4528f61009ae39642f5cfe4083874a857da"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L4ZcBn8xfLpTte7WKrbpcXW/7CRamb8+gvJv6tWZCGzhGDcRqmktlnHWo46Vk+0In1VvnC/1mrMGgEX0RA3xBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T14:50:14.391678Z","bundle_sha256":"23306248630b55e3d6f48dbf8c9c87b8f45e8d74460a1de167225808dc56f4b3"}}