Operator Algebra Generated by an Element from the Module \(\mathcal {B}(V_1,V_2)\)

  • Bogdan D. DjordjevićEmail author


Let \(B\in \mathcal {B}(V_1)\), \(A\in \mathcal {B}(V_2)\) and \(X\in \mathcal {B}(V_1,V_2)\) be given operators. In this paper the algebra \(\mathcal {A}_{AXB}\) generated by AXB is defined. Some results are obtained regarding invertibility in \(\mathcal {A}_{AXB}\). Special cases and extensions are discussed using algebraic representations. Applications to the operator equations \(AX=C\), \(XB=C\), \(AXB=C\), \(X-AXB=C\) and \(AX-XB=C\) are illustrated.


Operator algebras Operator modules Operator equations Topological function theory 

Mathematics Subject Classification

Primary 46H25 47L15 Secondary 30G12 



The author would like to thank the reviewer for carefully reading the manuscript and for the constructive comments which helped improve quality of the paper.


  1. 1.
    Bhatia, R.: Matrix Analysis. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bhatia, R., Rosenthal, P.: How and why to solve the operator equation \(AX-XB=Y\). Bull. Lond. Math. Soc. 29, 1–21 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chō, M., Načevska Nastovska, B.: Spectral properties of \(n-\)normal operators. Filomat 32(14), (2018).
  4. 4.
    Chō, M., Načevska Nastovska, B., Tomiyama, J.: On skew \([m, C]\)-symmetric operators. Adv. Oper. Theory 2(4), 468–474 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cojuhari, P.A.: Estimates of the discreete spectrum of a linear operator pencil. J. Math. Anal. Appl. 326, 1394–1409 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Contino, M., Giribet, J., Maestripieri, A.: Weighted least squares solutions of the equation \(AXB - C = 0\). Linear Algebra Appl. 518, 177–197 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Djordjević, B.D., Dinčić, N. Č.: Solving the operator equation \(AX-XB=C\) with closed \(A\) and \(B\). Integral Equ. Oper. Theory (2018).
  8. 8.
    Djordjević, D.S., Rakočević, V.: Lectures on Generalized Inverses. Faculty of Sciences and Mathematics, University of Niš, Niš (2008)zbMATHGoogle Scholar
  9. 9.
    Drazin, M.P.: On a result of J. J. Sylvester. Linear Algebra Appl. 505, 361–366 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eskandari, R., Fang, X., Sal Moslehian, M., Hu, Q.: Positive solutions of the system of operator equations \(A_1X=C_1\), \(XA_2=C_2\), \(A_3XA_3^*=C_3\) and \(A_4XA_4^*=C_4\) in Hilbert \(C^*-\)modules. Electron. J. Linear Algebra 34, 381–388 (2018)MathSciNetGoogle Scholar
  11. 11.
    Fang, X., Yu, J.: Solutions to operator equations on Hilbert \(C^*\)-modules 2. Integral Equ. Oper. Theory 68, 23–60 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gouveia, M.C.: On the solution of Sylvester, Lyapunov and Stein equations over arbitrary rings. Int. J. Pure Appl. Math. 24(1), 131–137 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hu, Q., Cheng, D.: The polynomial solution to the Sylvster matrix equation. Appl. Math. Lett. 19(9), 859–864 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jiang, T., Wei, M.: On solutions of the matrix equations \(X-AXB=C\) and \(X-A\overline{X}B=C\). Linear Algebra Appl. 367, 225–233 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Laura Arias, M., Celeste Gonzalez, M.: Positive solutions to operator equations \(AXB = C\). Linear Algebra Appl. 433, 1194–1202 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Laura Arias, M., Corach, G., Celeste Gonzalez, M.: Generalized inverses and Douglas equations. Proc. Am. Math. Soc. 136(9), 3177–3183 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mashreghi, J., Ransford, T.: Outer functions and divergence in de Branges–Rovnyak spaces. Complex Anal. Oper. Theory 12, 987–995 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Messirdi, B., Gherbi, A., Amouch, M.: A spectral analysis of linear operator pencils on Banach spaces with application to quotient of bounded operators. Int. J. Anal. Appl. 7(2), 104–128 (2015)zbMATHGoogle Scholar
  19. 19.
    Olteanu, O.: Invariant subspaces and invariant balls of bounded linear operators. Bull. Math. Soc. Sci. Math. Roum. Tome 59 (107), 261–271 (2016)Google Scholar
  20. 20.
    Pietsch, A.: Operator Ideals. North-Holland Publishing Company, Amsterdam (1980)zbMATHGoogle Scholar
  21. 21.
    Rashid, M.H.M., Tanahashi, K.: On commutator of generalized Aluthge transformations and Fuglede–Putnam theorem. Positivity 22(5), 1281–1295 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rosenblum, M.: On the operator equation \(AX-XB=Q\). Duke Math. J. 23, 263–270 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sylvester, J.J.: Sur l’equation en matrices \(px=xq\). C.R. Acad. Sci. Paris 99 (1884) 67–71 and 115–116Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences and MathematicsUniversity of NišNišRepublic of Serbia

Personalised recommendations