Growth of semigroups in discrete and continuous time

Alexander Gomilko, Hans Zwart, and Niels Besseling. "Growth of semigroups in discrete and continuous time." Studia Mathematica 206.3 (2011): 273-292. <http://eudml.org/doc/285379>.

@article{AlexanderGomilko2011,
abstract = {We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), $ẋ(t) = A^\{-1\} x(t)$, and the difference equation $x_\{d\}(n+1) = (A+I)(A-I)^\{-1\} x_\{d\}(n)$ are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup $(e^\{A^\{-1\}t\})_\{t≥0\}$ is O(∜t), and for $((A+I)(A-I)^\{-1\})ⁿ$ it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions on A such that the growth rate of $((A+I)(A-I)^\{-1\})ⁿ$ is O(1), i.e., the operator is power bounded.},
author = {Alexander Gomilko, Hans Zwart, Niels Besseling},
journal = {Studia Mathematica},
keywords = {Banach spaces; Hilbert spaces; -semigroups; generators; Cayley transform},
language = {eng},
number = {3},
pages = {273-292},
title = {Growth of semigroups in discrete and continuous time},
url = {http://eudml.org/doc/285379},
volume = {206},
year = {2011},
}

TY - JOUR
AU - Alexander Gomilko
AU - Hans Zwart
AU - Niels Besseling
TI - Growth of semigroups in discrete and continuous time
JO - Studia Mathematica
PY - 2011
VL - 206
IS - 3
SP - 273
EP - 292
AB - We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), $ẋ(t) = A^{-1} x(t)$, and the difference equation $x_{d}(n+1) = (A+I)(A-I)^{-1} x_{d}(n)$ are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup $(e^{A^{-1}t})_{t≥0}$ is O(∜t), and for $((A+I)(A-I)^{-1})ⁿ$ it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions on A such that the growth rate of $((A+I)(A-I)^{-1})ⁿ$ is O(1), i.e., the operator is power bounded.
LA - eng
KW - Banach spaces; Hilbert spaces; -semigroups; generators; Cayley transform
UR - http://eudml.org/doc/285379
ER -