{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:XQEUQSM6WIV3626QJFA5CAOPD7","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":"d9c96614ce2dd06fff3fccd46247dffd4e3d3a8ef74b47c08abd213228b5fffc","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2008-07-21T15:33:35Z","title_canon_sha256":"38f103b74206525949265c56e40459538aab44ece28c5787e7a3859e48c9e44a"},"schema_version":"1.0","source":{"id":"0807.3136","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0807.3136","created_at":"2026-05-18T00:14:23Z"},{"alias_kind":"arxiv_version","alias_value":"0807.3136v2","created_at":"2026-05-18T00:14:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.3136","created_at":"2026-05-18T00:14:23Z"},{"alias_kind":"pith_short_12","alias_value":"XQEUQSM6WIV3","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"XQEUQSM6WIV3626Q","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"XQEUQSM6","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:861845d85eb58e4a097c39069f12a0ab444cb0c6b219134f8422d7bcbe17a0e2","target":"graph","created_at":"2026-05-18T00:14:23Z","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 prove that if $n$ closed disks $D_1, D_2, ..., D_n$, of the Riemann sphere are spectral sets for a bounded linear operator $A$ on a Hilbert space, then their intersection $D_1\\cap D_2...\\cap D_n$ is a complete $K$-spectral set for $A$, with $K\\leq n+n(n-1)/\\sqrt3$. When $n=2$ and the intersection $X_1\\cap X_2$ is an annulus, this result gives a positive answer to a question of A.L. Shields (1974).","authors_text":"Bernhard Beckermann, Catalin Badea, Michel Crouzeix","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2008-07-21T15:33:35Z","title":"Intersections of several disks of the Riemann sphere as K-spectral sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.3136","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":"sha256:42d93f83f8d93e15afcc637adf34a357f3c84cfae9186e5ac6269b4762cebaa4","target":"record","created_at":"2026-05-18T00:14:23Z","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":"d9c96614ce2dd06fff3fccd46247dffd4e3d3a8ef74b47c08abd213228b5fffc","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2008-07-21T15:33:35Z","title_canon_sha256":"38f103b74206525949265c56e40459538aab44ece28c5787e7a3859e48c9e44a"},"schema_version":"1.0","source":{"id":"0807.3136","kind":"arxiv","version":2}},"canonical_sha256":"bc0948499eb22bbf6bd04941d101cf1fe9b0ae37de2f8f5260dfd6b3c414a4db","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc0948499eb22bbf6bd04941d101cf1fe9b0ae37de2f8f5260dfd6b3c414a4db","first_computed_at":"2026-05-18T00:14:23.457494Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:23.457494Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L9j4/YPY+YG9ijTZ4fzdYtSEt4mHvEtH/FSsBJFSZc6hctvZyDkzIZCS51/wk+mNb/OBLim+NLhR1HkWe6T6Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:23.457966Z","signed_message":"canonical_sha256_bytes"},"source_id":"0807.3136","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42d93f83f8d93e15afcc637adf34a357f3c84cfae9186e5ac6269b4762cebaa4","sha256:861845d85eb58e4a097c39069f12a0ab444cb0c6b219134f8422d7bcbe17a0e2"],"state_sha256":"9902932704addccd7fb8a5bbbbe8a420142b9915de309293fe62b83bf780d229"}