Continuity and invariance of the Sacker-Sell spectrum (Q300370)

This paper is devoted to the study of the exponential dichotomy (ED) spectrum of the linear nonautonomous difference equation \[ x(t+1)=A(t)x(t), \tag{1} \] where \(t\in \mathbb Z\), \(A:\mathbb Z\to [\mathbb R^{d}]\) and \([\mathbb R^d]\) is the space of all linear mapping acting on \(\mathbb R^{d}\). The dichotomy spectrum \(\Sigma(A)\) of (1) is defined as follows: \[ \Sigma(A):=\{\gamma >0 :\;x(t+1)=\gamma^{-1}A(t)x(t)\text{ has no ED}\}. \] The authors introduce some subsets of \(\Sigma(A)\): {\parindent=0.7cm \begin{itemize}\item[--] \(\Sigma_{s}(A):=\{\gamma >0:\;S_{\gamma} \;\text{is not onto}\}\). Here \(S_{\lambda}\) is a linear bounded operator defined by \((S_{\lambda}\phi)(t):=\phi(t+1)-\lambda^{-1}A(t)\phi(t)\); \item[--] \(\Sigma_{F}(A):=\{\gamma >0:\;S_{\gamma} \;\text{is not Fredholm}\}\); \item[--] \(\Sigma_{F_0}(A):=\{\gamma\in \Sigma_{F}(A):\;\text{the index of}\;S_{\gamma}\;\text{is zero}\}\); \item[--] \(\Sigma_{\pi}(A):=\{\gamma >0 :\;S_{\gamma}\;\text{is not bounded below}\}\). \end{itemize}} These spectrum subsets play an important role in the study of the asymptotic behavior of solutions of (1). The results in the paper are new and interesting for experts in the domain of qualitative theory of nonautonomous difference equations.