Articles

SOLUTIONS TO THE SYSTEM OF OPERATOR EQUATIONS AXB=C=BXA

  • Xiao ZHANG ,
  • Guoxing JI
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  • School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119, China

Received date: 2017-08-09

  Revised date: 2018-01-23

  Online published: 2018-08-25

Supported by

This research is supported by the National Natural Science Foundation of China (11371233).

Abstract

In this paper, we present some necessary and sufficient conditions for the existence of solutions, hermitian solutions and positive solutions to the system of operator equations AXB=C=BXA in the setting of bounded linear operators on a Hilbert space. Moreover, we obtain the general forms of solutions, hermitian solutions and positive solutions to the system above.

Cite this article

Xiao ZHANG , Guoxing JI . SOLUTIONS TO THE SYSTEM OF OPERATOR EQUATIONS AXB=C=BXA[J]. Acta mathematica scientia, Series B, 2018 , 38(4) : 1143 -1150 . DOI: 10.1016/S0252-9602(18)30804-X

References

[1] Arias M L, Gonzalez M C. Positive solutions to operator equations AXB=C. Linear Algebra Appl, 2010, 433(6):1194-1202
[2] Douglas R G. On majorization, factorization and range inclusion of operators in Hilbert space. Proc Amer Math Soc, 1966, 17(2):413-415
[3] Cvetković D S. Re-nnd solutions of the matrix equation AXB=C. J Aust Math Soc, 2008, 84(1):63-72
[4] Deng C. On the solutions of operator equation CAX=C=XAC. J Math Anal Appl, 2013, 398(2):664-670
[5] Don F J. Henk. On the symmetric solutions of a linear matrix equation. Linear Algebra Appl, 1987, 93:1-7
[6] Wang Q W, Wu Z C. Common hermitian solutions to some operator equations on Hilbert C*-modules. Linear Algebra Appl, 2010, 432(12):3159-3171
[7] Xu Q, Sheng L, Gu Y. The solutions to some operator equation. Linear Algebra Appl, 2008, 429:1997-2024
[8] Xu Q. Common hermitian and positive solutions to the adjointable operator equations AX=C, XB=D. Linear Algebra Appl, 2008, 429(1):1-11
[9] Lu T. Some nonlinear operator equations and their approximate solutions. Acta Mathematica Scientia, 1982, 2(4):421-435
[10] Dajić A, Koliha J J. Positive solutions to the equations AX=C, XB=D for Hilbert space operators. J Math Anal Appl, 2007, 333(2):567-576
[11] Vosough M, Moslehian M S. Solutions of the system of operator equations AXB=B=BXA via *-order. Electronic J Linear Algabra, 2017, 32:172-183
[12] Nashed M Z. Inner, outer, and generalized inverses in Banach and Hilbert spaces. Numer Funct Anal Optim, 1987, 9(3/4):261-325
[13] Xu X M, Du H K, Fang X C. The supremum of linear operators for the *-order. Linear Algebra Appl, 2010, 433(11):2198-2207
[14] Khatri C G, Mitra S K. Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J Appl Math, 1976, 31(4):579-585
[15] Yan K, Fang X C. Common properties of the operator products in spectral theory. Ann Funct Anal, 2015, 6:60-69
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