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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-2-2,
author = {Roland Schnaubelt},
title = {Exponential and polynomial dichotomies of operator semigroups on Banach spaces},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {121-138},
zbl = {1106.47037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-2-2}
}
Roland Schnaubelt. Exponential and polynomial dichotomies of operator semigroups on Banach spaces. Studia Mathematica, Tome 173 (2006) pp. 121-138. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-2-2/