On an estimate in the subspace perturbation problem
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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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- 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).Google Scholar