Exponential and polynomial dichotomies of operator semigroups on Banach spaces

Roland Schnaubelt¹

¹ FB Mathematik und Informatik Martin-Luther-Universität 06099 Halle, Germany

Studia Mathematica, Tome 175 (2006) no. 2, pp. 121-138

Résumé

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.

Détail

DOI : 10.4064/sm175-2-2

Keywords: generate semigroup cdot banach space resolvent tau exists uniformly bounded tau mathbb there exists closed possibly unbounded projection commuting moreover decays exponentially infty range exists decays exponentially to infty kernel domain depends fourier type tau only polynomially bounded obtains similar result polynomial decay application study partial functional differential equation