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Fredholm complements of upper triangular operator matrices

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Abstract

For a given operator pair \((A,B)\in (B(H),B(K))\), we denote by \(M_C\) the operator acting on a complex infinite dimensional separable Hilbert space \(H\oplus K\) of the form \(M_C=\bigl ( {\begin{matrix} A&{}C\\ 0&{}B \\ \end{matrix}}\bigr )\). This paper focuses on the Fredholm complement problems of \(M_C\). Namely, via the operator pair (AB), we look for an operator \(C\in B(K,H)\) such that \(M_C\) is Fredholm of finite ascent with nonzero nullity. As an application, we initiate the concept of the property (C) as a variant of Weyl’s theorem. At last, the stability of property (C) for \(2\times 2\) upper triangular operator matrices is investigated by the virtue of the so-called entanglement spectra of the operator pair (AB).

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Acknowledgements

The authors would like to extend their sincere gratitude to Professor Cao Xiaohong for her support and help. Meanwhile, thank to referees and editors for their valuable suggestions, which have greatly helped to improve the readability of this paper.

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Correspondence to Lining Jiang.

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Communicated by Gelu Popescu.

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Qiu, S., Jiang, L. Fredholm complements of upper triangular operator matrices. Banach J. Math. Anal. 18, 30 (2024). https://doi.org/10.1007/s43037-024-00340-2

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