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Minimality of root vectors of operator functions analytic in an angle

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Abstract

We study the minimality of elementsx h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers μ k of operator functionsL(λ) analytic in an angle; we assume that operators act in a Hilbert space\(\mathfrak{H}\). In particular, we consider the case whereL(λ)=I+T(λ)тc, β>0,I is an identity operator,C is a completely continuous operator, ∥(I- λC)−1∥≤c for ¦argλ¦≥θ, 0<θ<π, the operator functionT(λ) is analytic, and ∥T(λ)∥c for ¦argλ¦<θ. It is proved that, in this case, there exists ρ>0 such that the system of vectorsC v x h,j,k is minimal in\(\mathfrak{H}\) for arbitrary positive ν<1+β, provided that ¦μk¦>ρ.

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    G. V. Radzievskii

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  1. G. V. Radzievskii
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.

This research was partially supported by the Ukrainian State Committee of Science and Technology.

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Radzievskii, G.V. Minimality of root vectors of operator functions analytic in an angle. Ukr Math J 46, 581–603 (1994). https://doi.org/10.1007/BF01058521

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  • DOI: https://doi.org/10.1007/BF01058521

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