Abstract
Let A be a bounded self-adjoint operator on a separable Hilbert space ℌ and ℌ0 ⊂ ℌ a closed invariant subspace of A. Assuming that ℌ0 is of codimension 1, we study the variation of the invariant subspace ℌ0 under bounded self-adjoint perturbations V of A that are off-diagonal with respect to the decomposition ℌ = ℌ0 ⊕ ℌ1. In particular, we prove the existence of a one-parameter family of dense non-closed invariant subspaces of the operator A + V provided that this operator has a nonempty singularly continuous spectrum. We show that such subspaces are related to non-closable densely defined solutions of the operator Riccati equation associated with generalized eigenfunctions corresponding to the singularly continuous spectrum of B.
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Dedicated to Israel Gohberg on the occasion of his 75th birthday
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© 2005 Birkhäuser Verlag Basel/Switzerland
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Kostrykin, V., Makarov, K.A. (2005). The Singularly Continuous Spectrum and Non-Closed Invariant Subspaces. In: Gohberg, I., et al. Recent Advances in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 160. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7398-9_14
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DOI: https://doi.org/10.1007/3-7643-7398-9_14
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7290-3
Online ISBN: 978-3-7643-7398-6
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