Abstract
Consider a functionL(λ) defined on an interval Δ of the real axis whose values are linear bounded selfadjoint operators in a Hilbert spaceH. A point λ0 ∈ Δ and a vector ϕ0 ∈H( ϕ0 ≠ 0) are called eigenvalue and eigenvector ofL(λ) ifL(λ) ifL(λ0) ϕ0 = 0. Supposing that the functionL′(λ) can be represented as an absolutely convergent Fourier integral, the interval Δ is sufficiently small and the derivative will be positive at some point, it has been proved that all the eigenvectors of the operator-functionL(λ) corresponding to the eigenvalues from the interval Δ form an unconditional basis in the subspace spanned by them.
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References
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Matsaev, V., Spigel, E. Conditions for eigenvectors of a selfadjoint operator-function to form a basis. Integr equ oper theory 17, 443–449 (1993). https://doi.org/10.1007/BF01200296
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DOI: https://doi.org/10.1007/BF01200296