Estimates of reachable sets of control systems with nonlinearity and parametric perturbations |
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Authors: | T. F. Filippova |
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Affiliation: | 1.Institute of Mathematics and Mechanics,Ural Branch of the Russian Academy of Sciences,Yekaterinburg,Russia;2.Ural Federal University,Yekaterinburg,Russia |
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Abstract: | In the Banach space L1(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L2(M, τ) is introduced, and its main properties are established. A convergence criterion in L2(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L1(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖1 ≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L1(M, τ) is complemented. The convergence in L2(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained. |
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