Spectrum for compact operators on Banach spaces (Q2414885)

The authors investigate a generalized notion of spectrum for a nonautonomous dynamics in an infinite-dimensional setting inspired by the corresponding notion introduced by \textit{R. J. Sacker} and \textit{G. R. Sell} [J. Differ. Equations 27, 320--358 (1978; Zbl 0372.34027)] in a finite-dimensional space. For a sequence $(A_m)_{m\in \mathbb{Z}}$ of compact linear operators acting on a Banach space, they define its spectrum as the set of all $a \in \mathbb{R}$ such that the sequence $(e^{-a}A_m)_{m\in \mathbb{Z}}$ does not admit any exponential dichotomy. They give a characterization of all possible spectra and explicit examples of sequences for which the spectrum takes a form not occurring in finite-dimensional spaces are discussed. Finally, they consider the case of a one-sided sequence of compact linear operators.