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The Kachurovskij spectrum of Lipschitz continuous nonlinear block operator matrices

Abstract

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.

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Acknowledgements

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).

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Correspondence to Deyu Wu.

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Communicated by Matjaz Omladic.

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Dong, X., Wu, D. The Kachurovskij spectrum of Lipschitz continuous nonlinear block operator matrices. Adv. Oper. Theory 6, 54 (2021). https://doi.org/10.1007/s43036-021-00149-y

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Keywords

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

Mathematics Subject Classification

  • 47J10
  • 47H30