Abstract
A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.
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Acknowledgments
The authors would like to thank the referee for his/her careful reading, valuable comments and especially for pointing out a direct proof of Lemma 3.1. The authors wish to thank Professor Youqing Ji and Bingzhe Hou for valuable suggestions concerning this paper. This research was supported by NNSF of China (Nos. 11401088, 11326102, 11271150) and the Fundamental Research Funds for the Central Universities (No.12QNJJ001).
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Communicated by Mihai Putinar.
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Li, C.G., Zhou, T.T. Orthogonality Property \(\mathcal {O}\) and Compact Perturbations. Complex Anal. Oper. Theory 10, 1741–1755 (2016). https://doi.org/10.1007/s11785-016-0561-4
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DOI: https://doi.org/10.1007/s11785-016-0561-4