Exponential and polynomial dichotomies of operator semigroups on Banach spaces

Roland Schnaubelt. "Exponential and polynomial dichotomies of operator semigroups on Banach spaces." Studia Mathematica 175.2 (2006): 121-138. <http://eudml.org/doc/284841>.

@article{RolandSchnaubelt2006,
abstract = {Let A generate a C₀-semigroup T(·) on a Banach space X such that the resolvent R(iτ,A) exists and is uniformly bounded for τ ∈ ℝ. We show that there exists a closed, possibly unbounded projection P on X commuting with T(t). Moreover, T(t)x decays exponentially as t → ∞ for x in the range of P and T(t)x exists and decays exponentially as t → -∞ for x in the kernel of P. The domain of P depends on the Fourier type of X. If R(iτ,A) is only polynomially bounded, one obtains a similar result with polynomial decay. As an application we study a partial functional differential equation.},
author = {Roland Schnaubelt},
journal = {Studia Mathematica},
keywords = {dichotomy; splitting projection; Fourier type; strongly continuous semigroup; Gearhart's theorem; Weis-Wrobel theorem; resolvent estimates; fractional powers; delay equation},
language = {eng},
number = {2},
pages = {121-138},
title = {Exponential and polynomial dichotomies of operator semigroups on Banach spaces},
url = {http://eudml.org/doc/284841},
volume = {175},
year = {2006},
}

TY - JOUR
AU - Roland Schnaubelt
TI - Exponential and polynomial dichotomies of operator semigroups on Banach spaces
JO - Studia Mathematica
PY - 2006
VL - 175
IS - 2
SP - 121
EP - 138
AB - Let A generate a C₀-semigroup T(·) on a Banach space X such that the resolvent R(iτ,A) exists and is uniformly bounded for τ ∈ ℝ. We show that there exists a closed, possibly unbounded projection P on X commuting with T(t). Moreover, T(t)x decays exponentially as t → ∞ for x in the range of P and T(t)x exists and decays exponentially as t → -∞ for x in the kernel of P. The domain of P depends on the Fourier type of X. If R(iτ,A) is only polynomially bounded, one obtains a similar result with polynomial decay. As an application we study a partial functional differential equation.
LA - eng
KW - dichotomy; splitting projection; Fourier type; strongly continuous semigroup; Gearhart's theorem; Weis-Wrobel theorem; resolvent estimates; fractional powers; delay equation
UR - http://eudml.org/doc/284841
ER -