@article{Gorbachuk_Gorbachuk_2006, title={On the behavior of orbits of uniformly stable semigroups at infinity}, volume={58}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/3443}, abstractNote={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.}, number={2}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Gorbachuk, V. I. and Gorbachuk, M. L.}, year={2006}, month={Feb.}, pages={148–159} }