On the exponential dichotomy of the solutions of countable systems of linear differential equations (Q1376642)

In the space \(M\), the set of bounded sequences of numbers with the norm \(|x|= \sup|x_n|\) is considered a countable system of linear differential equations: \[ {dx_s\over dt}= \sum^\infty_1 a_{sj}(t)x_j,\tag{1} \] where \(x= (x_1,x_2,\dots)\in M\), \(\sup|x_n|<\infty\), \(a_{sj}(t)\), \(s,j= 1,2,\dots\) are bounded and continuous functions of the variable \(t\in(-\infty, +\infty)\). A solution of the system (1) is said to have the property of exponential dichotomy in the interval \((-\infty,+\infty)\), if for some \(t\in(-\infty, +\infty)\) the space \(M\) can be represented by a direct sum of two closed subspaces \(M= M^+\oplus M^-\), which satisfies also other properties. The author studies the exponential dichotomy of the system (1) by considering the exponential dichotomy of its shortened system (that is the system of a finite number of differential equations). In the second paragraph, the author studies the exponential dichotomy that depends on the shortened systems. In the last paragraph, intitled ``The stability of the exponential dichotomy'', the following system is considered: \[ {dy_s\over dt}= \sum^\infty_{j= 1} a_{sj}(t) y_j+ F_s(t, y_1,y_2,\dots),\quad s=1,2,\dots \] and he proves the lemma: If the system (1) has exponential dichotomy on \((-\infty,+ \infty)\), then there exists a transformation \(x= U(t)y\) which transforms (1) into a diagonal block system with \[ |U(t)|\leq K,\;|U^{-1}(t)|\leq K,\;\Biggl|{dU\over dt}\Biggr|\leq K, \] where \(K\) is a positive constant.