Lee, Sang-Gu; Vu, Quoc-Phong Simultaneous solutions of operator Sylvester equations. (English) Zbl 1302.47028 Stud. Math. 222, No. 1, 87-96 (2014). 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. Cited in 1 Document 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 Keywords:Sylvester equation; idempotent theorem; commutant; bicommutant; joint spectrum PDFBibTeX XMLCite \textit{S.-G. Lee} and \textit{Q.-P. Vu}, Stud. Math. 222, No. 1, 87--96 (2014; Zbl 1302.47028) Full Text: DOI