{"paper":{"title":"Strongly Integrable Operator-Valued Functions, Generated Vector Measures and Compactness of Integrals","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Every strongly integrable operator-valued function generates a countably additive B(X,Y)-valued measure in the operator norm when X* contains no copy of c0.","cross_cats":[],"primary_cat":"math.FA","authors_text":"Matija Milovi\\'c, Mihailo Krsti\\'c, Milo\\v{s} Arsenovi\\'c, Stefan Milo\\v{s}evi\\'c","submitted_at":"2026-05-12T17:43:58Z","abstract_excerpt":"Gel'fand integral of a family of compact operators on a Hilbert space is not always compact, even with additional property of positivity and commutativity.\n  We prove that integrals of a family, consisting of compact operators, in the space $L_{s}^1(\\Omega,\\mu,\\mathcal{B}(X, Y))$ of strongly integrable families are compact whenever $X$ does not contain an isomorphic copy of $\\ell^1$.\n  In addition, we prove an integral inequality for spectral radius $$r\\left(\\int_\\Omega\\mathscr{A} \\,d\\mu\\right)\\leqslant\\int_\\Omega r(\\mathscr{A}_t)\\,d\\mu(t)$$ for a mutually commuting family $\\mathscr{A}$ in $L_"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"every function in L_s^1(Ω,μ, B(X,Y)) generates a countably additive, in operator norm, B(X,Y)-valued measure whenever X* does not contain an isomorphic copy of c0","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that X* does not contain an isomorphic copy of c0 is load-bearing for the key theorem on generating countably additive measures; without it the measure may fail to be countably additive in norm, and the downstream compactness and spectral results would not follow.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Strongly integrable operator-valued functions generate norm-countably additive measures when X* avoids c0, so their integrals of compact operators are compact when X avoids ℓ1, plus a spectral radius inequality for commuting families.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every strongly integrable operator-valued function generates a countably additive B(X,Y)-valued measure in the operator norm when X* contains no copy of c0.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c859e4bd1a176f6fd3b2014f243af69e03eb377ea5fe321d3afe2dcd44214a0d"},"source":{"id":"2605.12454","kind":"arxiv","version":2},"verdict":{"id":"505586fb-69e1-4531-8c22-73398514f11c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T02:45:26.810756Z","strongest_claim":"every function in L_s^1(Ω,μ, B(X,Y)) generates a countably additive, in operator norm, B(X,Y)-valued measure whenever X* does not contain an isomorphic copy of c0","one_line_summary":"Strongly integrable operator-valued functions generate norm-countably additive measures when X* avoids c0, so their integrals of compact operators are compact when X avoids ℓ1, plus a spectral radius inequality for commuting families.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that X* does not contain an isomorphic copy of c0 is load-bearing for the key theorem on generating countably additive measures; without it the measure may fail to be countably additive in norm, and the downstream compactness and spectral results would not follow.","pith_extraction_headline":"Every strongly integrable operator-valued function generates a countably additive B(X,Y)-valued measure in the operator norm when X* contains no copy of c0."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.12454/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-19T22:21:57.828081Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T10:35:01.902204Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T08:01:18.200128Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T07:27:18.428604Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"14456adb94606739b1acfd9e409f20027d498abc0c46b5805682bc32457eddc3"},"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"}