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Joint spectrum of commuting self-adjoint operators and tests for propriety and stability for differential-operator equations

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Abstract

Tests are established for the propriety of the Cauchy problem, the stability, stabilization, asymptotic stability, and exponential stability for the equation y′'+Ay′+By=0, t ε [0, +∞), where A and B are arbitrary commuting self-adjoint operators on a separable Hilbert space. For this, in terms of the arrangement in R2 of the joint spectrum of A and B tests are obtained for B to be subordinate (strongly subordinate, equivalent) to A. The results on propriety and stability are illustrated by the example of model partial differential equations.

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Authors and Affiliations

  1. Institute of Mathematics of the Academy of Sciences of the UkrSSR, Kiev

    A. Ya. Shklyar

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  1. A. Ya. Shklyar
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 406–414, March, 1991.

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Shklyar, A.Y. Joint spectrum of commuting self-adjoint operators and tests for propriety and stability for differential-operator equations. Ukr Math J 43, 370–378 (1991). https://doi.org/10.1007/BF01670079

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  • DOI: https://doi.org/10.1007/BF01670079

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