Exponential trichotomy and qualitative properties for ordinary differential equations (Q2910252)

Consider a homogeneous linear system of differential equations, NEWLINE\[NEWLINE x'=A(t)x, \tag{1}NEWLINE\]NEWLINE where the matrix \(A\) is continuous on the real line.NEWLINENEWLINEThe authors say that system (1) admits an exponential trichotomy if it admits an exponential dichotomy on the positive half-line (with projection \(P\)), an exponential dichotomy on the negative half-line (with projection \(N\)), and \(PN=NP=N\).NEWLINENEWLINEThey show, for example, that if \(A\) is an almost automorphic function, then system (1) has an exponential trichotomy if and only it has an exponential dichotomy.NEWLINENEWLINEA pair of functional spaces \((E,F)\) is called admissible for system (1) if the system NEWLINE\[NEWLINE x'=A(t)x+f(t) NEWLINE\]NEWLINE has a unique solution \(x\in E\) for any \(f\in F\). Relations between exponential dichotomy and admissibility for some pairs of spaces are studied.