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Property \(\mathcal {P}\) and Compact Perturbations

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Let \(\mathcal {H}\) be a complex separable infinite dimensional Hilbert space. An operator T acting on \(\mathcal {H}\) is said to have property \(\mathcal {P}\), if \(\sigma (T)=\sigma _p(T)\) and \(\sigma (T^*)=\sigma _p(T^*)\). In this paper, we characterize those operators which have an arbitrarily small compact perturbation to satisfy property \(\mathcal {P}\). Also, we study the stability of property \(\mathcal {P}\) under small compact perturbations.

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Acknowledgements

The authors wish to thank Professor Chunguang Li for valuable suggestions concerning this paper. This work was supported by the National Natural Science Foundation of China No. 11901230.

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Authors and Affiliations

  1. Department of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Ting Ting Zhou

  2. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, People’s Republic of China

    Li Qi Xiu

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  1. Ting Ting Zhou
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All authors wrote the main manuscript text and all authors reviewed the manuscript.

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This article is part of the topical collection “Linear Operators and Linear Systems” edited by Sanne ter Horst, Dmitry S. Kaliuzhnyi-Verbovetskyi and Izchak Lewkowicz.

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Zhou, T.T., Xiu, L.Q. Property \(\mathcal {P}\) and Compact Perturbations. Complex Anal. Oper. Theory 18, 31 (2024). https://doi.org/10.1007/s11785-023-01450-9

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