Abstract
The problem of variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. A constrained optimization problem on a specific set of parameters is formulated, whose solution yields an estimate of the arcsine of the norm of the difference of the corresponding spectral projections. The solution is computed explicitly. The corresponding result is stronger than the one obtained by Albeverio and Motovilov and, in fact, is the best possible obtainable using their approach.
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References
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover Publications, New York, 1993.
S. Albeverio and A. K. Motovilov, Sharpening the norm bound in the subspace perturbation theory, Complex Anal. Oper. Theory 7 (2013), 1389–1416.
L. G. Brown, The rectifiable metric on the set of closed subspaces of Hilbert space, Trans. Amer. Math. Soc. 337 (1993), 279–289.
C. Davis, The rotation of eigenvectors by a perturbation, J. Math. Anal. Appl. 6 (1963), 159–173.
C. Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal. 7 (1970), 1–46.
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
V. Kostrykin, K. A. Makarov, and A. K. Motovilov, Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach, in Advances in Differential Equations and Mathematical Physics, Amer. Math. Soc., Providence, RI, 2003, pp. 181–198.
V. Kostrykin, K. A. Makarov, and A. K. Motovilov, On a subspace perturbation problem, Proc. Amer. Math. Soc. 131 (2003), 3469–3476.
V. Kostrykin, K. A. Makarov, A. K. Motovilov, Perturbation of spectra and spectral subspaces, Trans. Amer. Math. Soc. 359 (2007), 77–89.
K. A. Makarov and A. Seelmann, Metric properties of the set of orthogonal projections and their applications to operator perturbation theory, arXiv:1007.1575 [math.SP] (2010).
K. A. Makarov and A. Seelmann, The length metric on the set of orthogonal projections and new estimates in the subspace perturbation problem, J. Reine Angew. Math. 708 (2015), 1–15.
A. Seelmann, Notes on the sin 2 theorem, Integral Equations Operator Theory 79 (2014), 579–597.
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The material presented in this work is part of the author’s Ph.D. thesis.
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Seelmann, A. On an estimate in the subspace perturbation problem. JAMA 135, 313–343 (2018). https://doi.org/10.1007/s11854-018-0042-y
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DOI: https://doi.org/10.1007/s11854-018-0042-y