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Simultaneous solutions of operator Sylvester equations. (English) Zbl 1302.47028

Summary: We consider simultaneous solutions of operator Sylvester equations \(A_iX-XB_i=C_i \;(1\leq i \leq k)\), where \((A_1,\ldots ,A_k)\) and \((B_1,\ldots ,B_k)\) are commuting \(k\)-tuples of bounded linear operators on Banach spaces \({\mathcal E}\) and \({\mathcal F}\), respectively, and \((C_1,\ldots ,C_k)\) is a (compatible) \(k\)-tuple of bounded linear operators from \({\mathcal F}\) to \({\mathcal E}\), and prove that if the joint Taylor spectra of \((A_1,\ldots ,A_k)\) and \((B_1,\ldots ,B_k)\) do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

MSC:

47A62 Equations involving linear operators, with operator unknowns
47A10 Spectrum, resolvent
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
15A24 Matrix equations and identities
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