2017
Том 69
№ 6

All Issues

On the behavior of orbits of uniformly stable semigroups at infinity

Gorbachuk M. L., Gorbachuk V. I.

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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 $T (t)x, x ∈ B$, 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 $\{T(t)\}_{t \geq 0}$. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than $e^{-at^{\alpha}}$, where $a > 0,\; 0 < \alpha < \pi/(2 (\pi - 0 )),\; \theta$ is the angle of analyticity of $\{T(t)\}_{t \geq 0}$, and the collection of these orbits is dense in the set of all orbits.

English version (Springer): Ukrainian Mathematical Journal 58 (2006), no. 2, pp 163-177.

Citation Example: Gorbachuk M. L., Gorbachuk V. I. On the behavior of orbits of uniformly stable semigroups at infinity // Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 148–159.

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