Simultaneous solutions of operator Sylvester equations

Sang-Gu Lee ¹   ; Quoc-Phong Vu ²

¹ Department of Mathematics Sungkyunkwan University Suwon 440-746, Korea
² Department of Mathematics Ohio University Athens, OH 45701, USA and Vietnam Institute of Advanced Study in Mathematics Hanoi, Vietnam

Studia Mathematica, Tome 222 (2014) no. 1, pp. 87-96

Résumé

We consider simultaneous solutions of operator Sylvester equations $A_iX-XB_i=C_i \ (1\le i \le k)$, 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.

Détail

DOI : 10.4064/sm222-1-6

Keywords: consider simultaneous solutions operator sylvester equations ix xb where ldots ldots commuting k tuples bounded linear operators banach spaces mathcal mathcal respectively ldots compatible k tuple bounded linear operators mathcal mathcal prove joint taylor spectra ldots ldots intersect system sylvester equations has unique simultaneous solution