A+ CATEGORY SCIENTIFIC UNIT

Exponential and polynomial dichotomies of operator semigroups on Banach spaces

Volume 175 / 2006

Roland Schnaubelt Studia Mathematica 175 (2006), 121-138 MSC: Primary 47D06. DOI: 10.4064/sm175-2-2

Abstract

Let $A$ generate a $C_0$-semigroup $T(\cdot)$ on a Banach space $X$ such that the resolvent $R(i\tau,A)$ exists and is uniformly bounded for $\tau\in{\mathbb R}$. We show that there exists a closed, possibly unbounded projection $P$ on $X$ commuting with $T(t)$. Moreover, $T(t)x$ decays exponentially as $t\to\infty$ for $x$ in the range of $P$ and $T(t)x$ exists and decays exponentially as $t\to-\infty$ for $x$ in the kernel of $P$. The domain of $P$ depends on the Fourier type of $X$. If $R(i\tau,A)$ is only polynomially bounded, one obtains a similar result with polynomial decay. As an application we study a partial functional differential equation.

Authors

  • Roland SchnaubeltFB Mathematik und Informatik
    Martin-Luther-Universität
    06099 Halle, Germany
    e-mail

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