Dynamical spectrum for time dependent linear systems in Banach spaces (Q1343497)

The authors study a linear skew-product semiflow \(\pi\) defined on the phase space \({\mathcal E}= X\times \Theta\), where \(X\) is a Banach space and \(\Theta\) is a compact Hausdorff space. A notion of the exponential dichotomy for \(\pi\) is introduced. Finite dimension of the unstable subspace (lying in the fiber \(X_ \theta\)) is an essential feature of this notion. Then \(\lambda\in R\) belongs to the resolvent set \(\rho\in R\) of \(\pi\) iff the shifted semiflow \(\pi_ \lambda\) admits an exponential dichotomy. The structure of the dynamical spectrum \(\Sigma= R\backslash \rho\) (considered together with the corresponding spectral decomposition of \(\mathcal E\)) as well as the relationship between \(\Sigma\) and the Lyapunov exponents are established. The results of the paper extend some Sacker-Sell's theorem to the infinite-dimensional case. Examples of linear skew-product semiflows appearing in the case \(\dim X= \infty\) are given.