Pith Number

pith:YFE33M6C

pith:2026:YFE33M6CJEPHKVQS43JB5356N4

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Solving the Sylvester equation in Banach modules

Bogdan Djordjevi\'c

The Sylvester equation ax - xb = c is solvable in a Banach module precisely when c satisfies verifiable spectral compatibility conditions.

arxiv:2605.13419 v1 · 2026-05-13 · math.FA · math.OA

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Claims

C1strongest 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.

C2weakest 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.

C3one 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

References

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[1] Antoine, J.-P., Inoue, A., Trapani, C.:Partial∗−Algebras and their Oper- ator Realizations. Kluwer, Dordrecht (2002) 2002

[2] 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 1994

[3] 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) 2024

[4] 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 2026 · doi:10.1007/s43037-025-00465-y

[5] 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) 2024

Receipt and verification

First computed	2026-05-18T02:44:47.350276Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c149bdb3c2491e755612e6d21eefbe6f1aa31fc1fa510944341de02399278d96

Aliases

arxiv: 2605.13419 · arxiv_version: 2605.13419v1 · doi: 10.48550/arxiv.2605.13419 · pith_short_12: YFE33M6CJEPH · pith_short_16: YFE33M6CJEPHKVQS · pith_short_8: YFE33M6C

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/YFE33M6CJEPHKVQS43JB5356N4 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: c149bdb3c2491e755612e6d21eefbe6f1aa31fc1fa510944341de02399278d96

Canonical record JSON

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    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.FA",
    "submitted_at": "2026-05-13T12:14:17Z",
    "title_canon_sha256": "be58ae6d582c93919243789d2e1e6fabcada1b75a41d7edcd970f9930e0c7ac6"
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