Publishing house / Journals and Serials / Studia Mathematica / All issues

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.