On the behavior of orbits of uniformly stable semigroups at infinity

  • V. I. Gorbachuk
  • M. L. Gorbachuk

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.
Published
25.02.2006
How to Cite
Gorbachuk, V. I., and M. L. Gorbachuk. “On the Behavior of Orbits of Uniformly Stable Semigroups at Infinity”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, no. 2, Feb. 2006, pp. 148–159, http://umj.imath.kiev.ua/index.php/umj/article/view/3443.
Section
Research articles