{"paper":{"title":"Solving the Sylvester equation in Banach modules","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The Sylvester equation ax - xb = c is solvable in a Banach module precisely when c satisfies verifiable spectral compatibility conditions.","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Bogdan Djordjevi\\'c","submitted_at":"2026-05-13T12:14:17Z","abstract_excerpt":"For given unital complex Banach algebras $\\mathcal{A}_1$ and $\\mathcal{A}_2$, let $\\mathfrak{M}$ be a Banach module acting between them. Let $a\\in \\mathcal{A}_1$, $b\\in\\mathcal{A}_2$, and $c\\in\\mathfrak{M}$ be provided such that $\\sigma_{\\mathcal{A}_1}(a)\\cap\\sigma_{\\mathcal{A}_2}(b) \\neq\\emptyset$. In this paper we completely characterize the consistency of the Sylvester equation $$ax-xb=c.$$ Precisely, we establish verifiable sufficient and necessary solvability conditions, and we provide some formulas for particular solutions $x\\in\\mathfrak{M}$ when the equation is solvable. Moreover, we ch"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We completely characterize the consistency of the Sylvester equation ax-xb=c. Precisely, we establish verifiable sufficient and necessary solvability conditions, and we provide some formulas for particular solutions x∈M when the equation is solvable. Moreover, we characterize the uniqueness of the solutions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The setup assumes unital complex Banach algebras and a Banach module with continuous actions; if the module is not complete or the algebras lack units the spectral intersection condition and the derived solvability criteria may fail to apply or require substantial reformulation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Sylvester equation ax - xb = c is solvable in the Banach module precisely when verifiable spectral conditions hold, with explicit formulas for solutions and a characterization of uniqueness","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Sylvester equation ax - xb = c is solvable in a Banach module precisely when c satisfies verifiable spectral compatibility conditions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"463b9fbab1f645b89e137505408d5c7e4d431af52fbf6b219b5b4f1d13fd4bcb"},"source":{"id":"2605.13419","kind":"arxiv","version":1},"verdict":{"id":"6fde2743-c421-4d5d-8a46-0a8629173bb5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:20:14.718536Z","strongest_claim":"We completely characterize the consistency of the Sylvester equation ax-xb=c. Precisely, we establish verifiable sufficient and necessary solvability conditions, and we provide some formulas for particular solutions x∈M when the equation is solvable. Moreover, we characterize the uniqueness of the solutions.","one_line_summary":"The Sylvester equation ax - xb = c is solvable in the Banach module precisely when verifiable spectral conditions hold, with explicit formulas for solutions and a characterization of uniqueness","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The setup assumes unital complex Banach algebras and a Banach module with continuous actions; if the module is not complete or the algebras lack units the spectral intersection condition and the derived solvability criteria may fail to apply or require substantial reformulation.","pith_extraction_headline":"The Sylvester equation ax - xb = c is solvable in a Banach module precisely when c satisfies verifiable spectral compatibility conditions."},"references":{"count":32,"sample":[{"doi":"","year":2002,"title":"Antoine, J.-P., Inoue, A., Trapani, C.:Partial∗−Algebras and their Oper- ator Realizations. Kluwer, Dordrecht (2002)","work_id":"3fa4cf47-cb61-4499-bb92-742d686f8e04","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"W. Arendt, F. R¨ abiger and A. Sourour,Spectral properties of the operator equationAX+XB=Y, Quart. J. Math. Oxford 2:45 (1994) 133–149","work_id":"a74f5174-b38d-4b46-8584-fa1fe143060c","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Bellomonte, G., Djordjevi´ c, B., Ivkovi´ c, S.,On representations and topo- logical aspects of positive maps on non-unital quasi∗−algebras, Positivity 28(5), 66 (2024)","work_id":"35bd9025-9d2b-4186-bc42-e4df2263be68","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s43037-025-00465-y","year":2026,"title":"Bellomonte, G. Ivkovi´ c, S. Trapani, Banach bimodule-valued positivemaps: inequalities and representations, Banach J. Math. Anal. (2026) 20:12. https://doi.org/10.1007/s43037-025-00465-y","work_id":"78804de9-7bf5-446e-92e5-344dd6d84e5f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Bellomonte, G. Ivkovi´ c, S. Trapani, C.,GNS construction for positive C ∗−valued sesquilinear maps on a quasi∗−aglebra, Mediterr. J. Math., 21 (2024) 166 (22 pp) (2024)","work_id":"f6371c6e-2545-44a3-b017-8125225257a8","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":32,"snapshot_sha256":"30069d7a388b899eeca105d41ec16705c8cad8cbdee9f3854b15e6005a9ec85a","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"}