{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1997:ZWO5ECV7IJXHTHIIWFYFHGNTKY","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":"8bed12c4ffacc42cb89308227fd74efde830d304adecadea055a7b6b16c66190","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1997-09-08T00:00:00Z","title_canon_sha256":"5fd9a177fc9888336fe3238058dd064aa7f4e0014a1209fd2cbd1673962301a6"},"schema_version":"1.0","source":{"id":"math/9709209","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9709209","created_at":"2026-05-18T01:05:34Z"},{"alias_kind":"arxiv_version","alias_value":"math/9709209v1","created_at":"2026-05-18T01:05:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9709209","created_at":"2026-05-18T01:05:34Z"},{"alias_kind":"pith_short_12","alias_value":"ZWO5ECV7IJXH","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_16","alias_value":"ZWO5ECV7IJXHTHII","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_8","alias_value":"ZWO5ECV7","created_at":"2026-05-18T12:25:49Z"}],"graph_snapshots":[{"event_id":"sha256:1b8b7c2574dfede1df75abb6eed2777c31fd31854a5ca57eaeab56fb7a8405d2","target":"graph","created_at":"2026-05-18T01:05:34Z","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":"Suppose $\\Cal J$ is a two-sided quasi-Banach ideal of compact operators on a separable infinite-dimensional Hilbert space $\\Cal H$. We show that an operator $T\\in\\Cal J$ can be expressed as finite linear combination of commutators $[A,B]$ where $A\\in\\Cal J$ and $B\\in\\Cal B(\\Cal H)$ if and only its eigenvalues $(\\lambda_n)$ (arranged in decreasing order of absolute value, repeated according to algebraic multiplicity and augmented by zeros if necessary) satisfy the condition that the diagonal operator $\\diag\\{\\frac1n(\\lambda_1+\\cdots +\\lambda_n)\\}$ is a member of $\\Cal J.$ This answers (for quas","authors_text":"Nigel J. Kalton","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"1997-09-08T00:00:00Z","title":"Spectral characterization of sums of commutators I"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9709209","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":"sha256:461aed0e25fb78435aafba97aaa001383362e86f4b7434960d01113811d32958","target":"record","created_at":"2026-05-18T01:05:34Z","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":"8bed12c4ffacc42cb89308227fd74efde830d304adecadea055a7b6b16c66190","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1997-09-08T00:00:00Z","title_canon_sha256":"5fd9a177fc9888336fe3238058dd064aa7f4e0014a1209fd2cbd1673962301a6"},"schema_version":"1.0","source":{"id":"math/9709209","kind":"arxiv","version":1}},"canonical_sha256":"cd9dd20abf426e799d08b1705399b3561000d262381f4bc01c4c603613023081","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd9dd20abf426e799d08b1705399b3561000d262381f4bc01c4c603613023081","first_computed_at":"2026-05-18T01:05:34.863148Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:34.863148Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o1FT0MwllMvwnRW4B1RYRdUgyS27GSNkgrrpK79lHOMIj/dRiKdST92oDGR0DZeO8Jp50g7xksYKk6lpEIfVCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:34.863862Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9709209","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:461aed0e25fb78435aafba97aaa001383362e86f4b7434960d01113811d32958","sha256:1b8b7c2574dfede1df75abb6eed2777c31fd31854a5ca57eaeab56fb7a8405d2"],"state_sha256":"e4b9f5de1d08e59126a16defda623ef3406b73c7fdbd9a9b0f58539d611a0344"}