Simultaneous solutions of operator Sylvester equations
Volume 222 / 2014
Studia Mathematica 222 (2014), 87-96
MSC: Primary 47A62, 47A10, 47A13; Secondary 15A24.
DOI: 10.4064/sm222-1-6
Abstract
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.
