On perturbation theory for points of discrete spectrum of dissipative operator functions |
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Authors: | A. A. Shkalikov |
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Affiliation: | (1) Moscow State University, Moscow, Russia |
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Abstract: | We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α 0 ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α 0 and R is a bounded positive operator; moreover, the point α 0 is a normal eigenvalue of the operator function L(α, θ 0) for some θ 0 ∈ ?, and the number θ 0 is a normal eigenvalue of the operator function L(α 0 θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α 0 and θ 0, respectively, and on the representation of the corresponding eigenfunctions by series. |
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