{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:XE7KYDMYY32ZNNI7ST5F2QIHI5","short_pith_number":"pith:XE7KYDMY","canonical_record":{"source":{"id":"1206.0492","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-03T21:34:52Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"8101a785fbd2f92ec0598004f35c227247f2b78291815bb1c75b54933e281bbb","abstract_canon_sha256":"8a7fdc527f4f2d9b7c71ba9327cc64e24cc791e221fffde5c75d96eda7bd61ee"},"schema_version":"1.0"},"canonical_sha256":"b93eac0d98c6f596b51f94fa5d410747648b41b93339084dcce4cb650b0d9595","source":{"kind":"arxiv","id":"1206.0492","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.0492","created_at":"2026-05-18T03:54:17Z"},{"alias_kind":"arxiv_version","alias_value":"1206.0492v1","created_at":"2026-05-18T03:54:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.0492","created_at":"2026-05-18T03:54:17Z"},{"alias_kind":"pith_short_12","alias_value":"XE7KYDMYY32Z","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XE7KYDMYY32ZNNI7","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XE7KYDMY","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:XE7KYDMYY32ZNNI7ST5F2QIHI5","target":"record","payload":{"canonical_record":{"source":{"id":"1206.0492","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-03T21:34:52Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"8101a785fbd2f92ec0598004f35c227247f2b78291815bb1c75b54933e281bbb","abstract_canon_sha256":"8a7fdc527f4f2d9b7c71ba9327cc64e24cc791e221fffde5c75d96eda7bd61ee"},"schema_version":"1.0"},"canonical_sha256":"b93eac0d98c6f596b51f94fa5d410747648b41b93339084dcce4cb650b0d9595","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:17.527894Z","signature_b64":"3InoOf95a9UIcxDQcL5vdChXGZyMwGkChe1iJa0+kTYCsr3g5ARy3sTRAx0zc70lTXLFdeecgRkB4M3RoMzmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b93eac0d98c6f596b51f94fa5d410747648b41b93339084dcce4cb650b0d9595","last_reissued_at":"2026-05-18T03:54:17.527129Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:17.527129Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.0492","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":"2026-05-18T03:54:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Ct0AzhD7CDtemo8Iuz5qOAG1oG1IuvkWfUkQjDKx0H/Cav/a85YfewcEA0q8mBW8Nzn0uppN654PxRARJcLDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T19:01:36.270043Z"},"content_sha256":"dfc6b50f241025deada6b7f982659ebc38c2589d0b0be867d75281b66ff74606","schema_version":"1.0","event_id":"sha256:dfc6b50f241025deada6b7f982659ebc38c2589d0b0be867d75281b66ff74606"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:XE7KYDMYY32ZNNI7ST5F2QIHI5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the power-bounded operators of classes $C_{0 \\cdot}$ and $C_{1 \\cdot}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Patryk Pagacz","submitted_at":"2012-06-03T21:34:52Z","abstract_excerpt":"By a bounded backward sequence of the operator $T$ we mean a bounded sequence $\\{x_n\\}$ satisfying $Tx_{n+1}=x_n$. In \\cite{Pa} we have characterized contractions with strongly stable nonunitary part in terms of bounded backward sequences. The main purpose of this work is to extend that result to power-bounded operators. Aditionally, we show that a power-bounded operator is strongly stable ($C_{0 \\cdot} $) if and only if its adjoint does not have any nonzero bounded backward sequence. Similarly, a power-bounded operator is non-vanishing ($C_{1 \\cdot} $) if and only if its adjoint has a lot of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0492","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":"2026-05-18T03:54:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tftCsmjfEO7y2nLgHXDRP0JWNN2TBaPr8TmAgjyoWhFnTUtfQb0hbztxRYXONs/8uaEcw1PkjSOjjUDDrRuNDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T19:01:36.270394Z"},"content_sha256":"84d8da6e02a78f7110b0d9a4f8a3943809cc075b925c1792bc608668d1db16ae","schema_version":"1.0","event_id":"sha256:84d8da6e02a78f7110b0d9a4f8a3943809cc075b925c1792bc608668d1db16ae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XE7KYDMYY32ZNNI7ST5F2QIHI5/bundle.json","state_url":"https://pith.science/pith/XE7KYDMYY32ZNNI7ST5F2QIHI5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XE7KYDMYY32ZNNI7ST5F2QIHI5/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":"2026-06-25T19:01:36Z","links":{"resolver":"https://pith.science/pith/XE7KYDMYY32ZNNI7ST5F2QIHI5","bundle":"https://pith.science/pith/XE7KYDMYY32ZNNI7ST5F2QIHI5/bundle.json","state":"https://pith.science/pith/XE7KYDMYY32ZNNI7ST5F2QIHI5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XE7KYDMYY32ZNNI7ST5F2QIHI5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XE7KYDMYY32ZNNI7ST5F2QIHI5","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":"8a7fdc527f4f2d9b7c71ba9327cc64e24cc791e221fffde5c75d96eda7bd61ee","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-03T21:34:52Z","title_canon_sha256":"8101a785fbd2f92ec0598004f35c227247f2b78291815bb1c75b54933e281bbb"},"schema_version":"1.0","source":{"id":"1206.0492","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.0492","created_at":"2026-05-18T03:54:17Z"},{"alias_kind":"arxiv_version","alias_value":"1206.0492v1","created_at":"2026-05-18T03:54:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.0492","created_at":"2026-05-18T03:54:17Z"},{"alias_kind":"pith_short_12","alias_value":"XE7KYDMYY32Z","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XE7KYDMYY32ZNNI7","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XE7KYDMY","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:84d8da6e02a78f7110b0d9a4f8a3943809cc075b925c1792bc608668d1db16ae","target":"graph","created_at":"2026-05-18T03:54:17Z","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":"By a bounded backward sequence of the operator $T$ we mean a bounded sequence $\\{x_n\\}$ satisfying $Tx_{n+1}=x_n$. In \\cite{Pa} we have characterized contractions with strongly stable nonunitary part in terms of bounded backward sequences. The main purpose of this work is to extend that result to power-bounded operators. Aditionally, we show that a power-bounded operator is strongly stable ($C_{0 \\cdot} $) if and only if its adjoint does not have any nonzero bounded backward sequence. Similarly, a power-bounded operator is non-vanishing ($C_{1 \\cdot} $) if and only if its adjoint has a lot of ","authors_text":"Patryk Pagacz","cross_cats":["math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-03T21:34:52Z","title":"On the power-bounded operators of classes $C_{0 \\cdot}$ and $C_{1 \\cdot}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0492","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:dfc6b50f241025deada6b7f982659ebc38c2589d0b0be867d75281b66ff74606","target":"record","created_at":"2026-05-18T03:54:17Z","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":"8a7fdc527f4f2d9b7c71ba9327cc64e24cc791e221fffde5c75d96eda7bd61ee","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-03T21:34:52Z","title_canon_sha256":"8101a785fbd2f92ec0598004f35c227247f2b78291815bb1c75b54933e281bbb"},"schema_version":"1.0","source":{"id":"1206.0492","kind":"arxiv","version":1}},"canonical_sha256":"b93eac0d98c6f596b51f94fa5d410747648b41b93339084dcce4cb650b0d9595","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b93eac0d98c6f596b51f94fa5d410747648b41b93339084dcce4cb650b0d9595","first_computed_at":"2026-05-18T03:54:17.527129Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:54:17.527129Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3InoOf95a9UIcxDQcL5vdChXGZyMwGkChe1iJa0+kTYCsr3g5ARy3sTRAx0zc70lTXLFdeecgRkB4M3RoMzmAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:54:17.527894Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.0492","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dfc6b50f241025deada6b7f982659ebc38c2589d0b0be867d75281b66ff74606","sha256:84d8da6e02a78f7110b0d9a4f8a3943809cc075b925c1792bc608668d1db16ae"],"state_sha256":"cefa425a00ca714d8ce349056a5c3055ad33c0453dcaaeedb575c36c7402baaa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ry7+iPjeJamGZZ5pGdF/DIvUGmIL4JYW/P28O93D3y7SbpPQGEdVfRTjWCs8iXBotlbW2aF/mNipfnoOf+akAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T19:01:36.272301Z","bundle_sha256":"c6e3b1f3a5e9908d91bf5ba4079af6f7399fafc7fe299a0826670fd8fa732b2c"}}