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On the behavior of orbits of uniformly stable semigroups at infinity

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Abstract

For uniformly stable bounded analytic C 0-semigroups {T(t)} t≥0 of linear operators in a Banach space B, we study the behavior of their orbits T (t)x, xB, at infinity. We also analyze the relationship between the order of approaching the orbit T (t)x to zero as t → ∞ and the degree of smoothness of the vector x with respect to the operator A −1 inverse to the generator A of the semigroup {T(t)} t≥0. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than \(e^{ - at^\alpha } \), where a > 0, 0 < α < π/(2(π-θ)), θ is the angle of analyticity of {T(t)} t≥0, and the collection of these orbits is dense in the set of all orbits.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 148–159, February, 2006.

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Horbachuk, V.I., Horbachuk, M.L. On the behavior of orbits of uniformly stable semigroups at infinity. Ukr Math J 58, 163–177 (2006). https://doi.org/10.1007/s11253-006-0059-8

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  • DOI: https://doi.org/10.1007/s11253-006-0059-8

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