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Simultaneous solutions of operator Sylvester equations

Volume 222 / 2014

Sang-Gu Lee, Quoc-Phong Vu 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.

Authors

  • Sang-Gu LeeDepartment of Mathematics
    Sungkyunkwan University
    Suwon 440-746, Korea
    e-mail
  • Quoc-Phong VuDepartment of Mathematics
    Ohio University
    Athens, OH 45701, USA
    and
    Vietnam Institute of Advanced Study in Mathematics
    Hanoi, Vietnam
    e-mail

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