The Kachurovskij spectrum of Lipschitz continuous nonlinear block operator matrices


In this paper, the Kachurovskij spectrum of \(2\times 2\) Lipschitz continuous nonlinear operator matrices are studied. Firstly, some connections between the Kachurovskij spectrum of certain \(2\times 2\) Lipschitz continuous nonlinear operator matrices and that of their entries are established, and the relationship between the Kachurovskij spectrum of \(2\times 2\) Lipschitz continuous nonlinear operator matrices and that of their Schur complement is presented by means of Schur decomposition. Then, the Gershgorin’s theorem of \(2\times 2\) Lipschitz continuous nonlinear operator matrices is given, and the spectral inclusion properties of Lipschitz continuous nonlinear block operator matrices are investigated by using the numerical range of diagonal entries. Finally, the lipeomorphism of certain \(2\times 2\) Lipschitz continuous nonlinear operator matrices is characterized by using the perturbation theory of nonlinear operators.

This is a preview of subscription content, access via your institution.


  1. 1.

    Amar, A., Jeribi, A., Krichen, B.: Fixed point theorems for block operator matrix and an application to a structured problem under boundary conditions of Rotenberg’s model type. Math. Slovaca 64, 155–174 (2014)

  2. 2.

    Apostol, C.: The reduced minimum modulus. Michigan Math. J. 32, 279–294 (1985)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Appell, J., Dörfner, M.: Some spectral theory for nonlinear operators. Nonlinear Anal. TMA. 28, 1955–1976 (1997)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Appell, J., Pascale, E.D., Vignoli, A.: Nonlinear Spectral Theory. Walter de Gruyter, Berlin (2004)

    Book  Google Scholar 

  5. 5.

    Barraa, M., Boumazgour, M.: A note on the spectrum of an upper triangular operator matrix. Proc. Amer. Math. Soc. 131, 3083–3088 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cao, X.H., Meng, B.: Essential approximate point spectra and Weyl’s theorem for operator matrices. J. Math. Anal. Appl. 304, 759–771 (2005)

  7. 7.

    Chen, A., Bai, Q.M., Wu, D.Y.: Spectra of \(2\times 2\) upper triangular operator matrices with unbounded entries (in Chinese). Sci. Sin. Math. (Chin. Ser.) 46, 157–168 (2016)

    Article  Google Scholar 

  8. 8.

    Chiappinelli, R.: Surjectivity of coercive gradient operators in Hilbert space and nonlinear spectral theory. Ann. Funct. Anal. 10, 1–9 (2018)

    MathSciNet  Google Scholar 

  9. 9.

    Clancey, K.: Seminormal Operators. Springer, Berlin (1979)

    Book  Google Scholar 

  10. 10.

    Halilovic, S., Sadikovic, S.: The Point and Rhodius spectra of certain nonlinear superposition operators. Adv. Math. 7, 1–8 (2018)

    MATH  Google Scholar 

  11. 11.

    Hashem, H.H.G.: Solvability of a \(2\times 2\) block operator matrix of chandrasekhar type on a Bananch algebra. Filomat 31, 5169–5175 (2017)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hashem, H., El-Sayed, A., Baleanu, D.: Existence results for block matrix operator of fractional orders in Banach algebras. Mathematics 7, 856 (2019)

    Article  Google Scholar 

  13. 13.

    Ize, J., Vignoli, A.: Equivariant nonlinear spectrum. J. Fixed Point Theory Appl. 13, 51–62 (2013)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Jeribi, A., Kaddachi, N., Krichen, B.: Fixed-point theorems for multivalued operator matrix under weak topology with an application. Bull. Malays. Math. Sci. Soc. 43, 1047–1067 (2020)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Kachurovskij, R.I.: Regular points, spectrum and eigenfunctions of nonlinear operators. Soviet Math. Dokl. 10, 1101–1105 (1969)

    MATH  Google Scholar 

  16. 16.

    Kaddachi, N., Jeribi, A., Krichen, B.: Fixed point theorems of block operator matrices on Banach algebras and an application to functional integral equations. Math. Meth. Appl. Sci. 36, 659–673 (2013)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Maddox, I.J., Wickstead, A.W.: The spectrum of uniformly Lipschitz mappings. Proc. Roy. Irish Acad. Sect. A. 89, 101–114 (1989)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Nagel, R.: Towards a “matrix theory” for unbounded operator matrices. Math. Z. 201, 57–68 (1989)

  19. 19.

    Nagel, R.: The spectral of unbounded operator matrices with non-diagonal domain. J. Funct. Anal. 89, 291–302 (1990)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Qi, Y.R., Huang, J.J., Chen, A.: Spectral inclusion properties of some unbounded block operator matrices (in Chinese). Sci. Sin. Math. (Chin. Ser.) 44, 1099–1110 (2014)

    Article  Google Scholar 

  21. 21.

    Salas, H.N.: Gershgorin’s theorem for matrices of operators. Linear Algebra Appl. 291, 15–36 (1999)

  22. 22.

    Toeplitz, O.: Das algebraische Analogon zu einem Satz von F\(\acute{e}\)j\(\grave{e}\)r. Math. Z. 2, 187–197 (1918)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Tretter, C.: Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London (2008)

    Book  Google Scholar 

  24. 24.

    Wang, L.S., Xu, Z.B.: Quantitative properties of nonlinear Lipschitz continuous operators. IV. Spectral theory (in Chinese). Acta Math Sinica (Chin. Ser.) 38, 628–631 (1995)

    Google Scholar 

  25. 25.

    Wu, X.F., Huang, J.J.: Essential spectrum of upper triangular operator matrices. Ann. Funct. Anal. 11, 780–798 (2020)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Zarantonello, E.H.: The closure of the numerical range contains the spectrum. Pac. J. Math. 22, 575–595 (1967)

    MathSciNet  Article  Google Scholar 

Download references


The authors are grateful to the referee and editor for their valuable comments and suggestions on this paper. The research is supported by the NNSF of China (Grant Nos. 11561048, 11761029), NSF of Inner Mongolia (Grant Nos. 2019MS01019, 2020ZD01) and the Postgraduate Scientific Research Innovation Foundation of Inner Mongolia University (Grant No. 11200-121024).

Author information



Corresponding author

Correspondence to Deyu Wu.

Additional information

Communicated by Matjaz Omladic.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dong, X., Wu, D. The Kachurovskij spectrum of Lipschitz continuous nonlinear block operator matrices. Adv. Oper. Theory 6, 54 (2021).

Download citation


  • Nonlinear operator matrices
  • Lipschitz continuous operator
  • Kachurovskij spectrum
  • Gershgorin theorem

Mathematics Subject Classification

  • 47J10
  • 47H30