Abstract
The spectral properties of two products AB and BA of possibly unbounded operators A and B in a Banach space are considered. The results are applied in the comparison of local spectral properties of the operators \({T^{[\ast]} T}\) and \({TT^{[\ast]}}\) in a Krein space. It is shown that under the assumption that both operators \({T^{[\ast]} T}\) and \({TT^{[\ast]}}\) have non-empty resolvent sets, the operator \({T^{[\ast]} T}\) is locally definitizable if and only if \({TT^{[\ast]}}\) is. In this context the critical points of both operators are compared.
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Acknowledgements
We would like to thank our colleagues Piotr Budzyński and Carsten Trunk for helpful discussions and inspirations. We also would like to thank the referee for his useful comments, which helped to improve the paper.
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The first author gratefully acknowledges support from the Deutsche Forschungsgemeinschaft (DFG), grant BE 3765/5-1 TR 904/4-1. The third author was supported by the EU Sixth Framework Programme for the Transfer of Knowledge “Operator theory methods for differential equations” (TODEQ) # MTKD-CT-2005-030042, he also thanks the Faculty of Sciences of the VU University Amsterdam, where the research was partially carried out during his Post-Doc stay.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Philipp, F., Ran, A.C.M. & Wojtylak, M. Local Definitizability of \({{T^{[\ast]}T}}\) and \({{TT^{{[\ast]}}}}\) . Integr. Equ. Oper. Theory 71, 491–508 (2011). https://doi.org/10.1007/s00020-011-1913-0
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DOI: https://doi.org/10.1007/s00020-011-1913-0