Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (6): 1640-1652.

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Solvability of Several Kinds of Ggeneralized Sylvester Operator Equations

Sun Xiaolin, Wang Hua   

  1. College of Sciences, Inner Mongolia University of Technology, Hohhot 010051
  • Received:2021-10-26 Revised:2022-04-20 Published:2022-12-16
  • Supported by:
    Supported by the NSFC (11961052), the NSF of Inner Mongolia (2022MS01005, 2021LHMS01004), the Basic Science Research Fund in the University Directly under the Inner Mongolia Autonomous Region (JY220083, JY20220151) and Tenching Reform of Postgraduate Education in Inner Mongolia University of Technology (YJG2020027)

Abstract: In this paper, we discuss the solvability of some kinds of generalized Sylvester operator equations on Hilbert Spaces. Firstly, the necessary and sufficient conditions for the existence of solutions of $AXB-XD=EB$ and $AXB-CX =AE$ are given in the case of normal operators. Secondly, the solvability of three kinds of operator equations is discussed by using the uniform equivalence of operator pairs. Finally, when trigonometric operator matrix is similar to diagonal operator matrix, the corresponding operator equations are solvable.

Key words: Sylvester equation, Operator equation, Normal operator, Putnam-Fuglede's theorem

CLC Number: 

  • O151.1