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Essential spectra of upper triangular relation matrices

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Abstract

Let H and K be infinite dimensional complex separable Hilbert spaces. Given the operators \(A\in \mathcal{L}\mathcal{R}(H)\), \(B\in \mathcal{L}\mathcal{R}(K)\) and \(C\in \mathcal{L}\mathcal{R}(K,H)\), we define upper triangular linear relation matrix \(M_{C}:=\left( {\begin{matrix} A &{}C \\ 0 &{} B \\ \end{matrix}} \right) \). In this paper, we obtain \(\sigma _{\star }(M_{C})\subseteq \sigma _{\star }(\left[ {\begin{matrix} A &{}C(0)\\ \end{matrix}} \right] _{H})\cup \sigma _{\star }(B),\) and the relations between \(\sigma _{\star }(M_{C})\) and \(\sigma _{\star }(A)\cup \sigma _{\star }(B)\) are also presented, where \(\sigma _{\star }\) is chosen from the essential spectrum, the Weyl spectrum, the essential approximate point spectrum, the Browder spectrum and the Browder essential approximate point spectrum.

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Acknowledgements

The authors are very grateful to the anonymous referees for their invaluable and instructive suggestions and comments which have greatly improved this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, China

    Yanyan Du

  2. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Junjie Huang

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  1. Yanyan Du
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  2. Junjie Huang
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Correspondence to Junjie Huang.

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Communicated by Gerald Teschl.

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This work is supported by the NNSF of China (No. 11961052), the NSF of Inner Mongolia (No. 2021MS01006), and the DSRF of Shandong University of Technology (No. 4041/421092).

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Du, Y., Huang, J. Essential spectra of upper triangular relation matrices. Monatsh Math 200, 43–61 (2023). https://doi.org/10.1007/s00605-022-01800-3

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