Reducibility of a linear system of differential equations with odd almost-periodic coefficients (Q1387298)

Consider the linear differential equation \(\dot X=A(t,z)X\) in \(\mathbb{C}^n\), where \(A(t,z)\) is a matrix almost-periodic in \(t\) of the form \[ A(t,z)=\sum_{k\geq 0} A_k(t)z^k,\qquad A_k(t)=\sum_{\| m\|\leq N_k}A_{k,m}e^{i(m,\omega)t}.\tag \(*\) \] With \(\| m\|=| m_1| +\dots+| m_{n_k}| \), the summation is over \(n_k\)-dimensional integer vectors \(m\); \(\{n_k\}\) and \(\{N_k\}\) are nondecreasing sequences of positive integers such that \(n_0=0\) and \(N_{k+l}\geq N_k+N_l\), \(\omega=(\omega_1,\omega_2,\dots)\) is a sequence of rationally independent real numbers, \((m,\omega)=m_1\omega_1+\dots+m_{n_k}\omega_{n_k}\), and \(A_{k,m}\) are bounded matrices. Let \(H_r\) be the space of operators of the form \((*)\) such that \[ \| A\|_r=\sup_{t\in\mathbb{R},| z| \leq r}\| A(t,z)\|<+\infty, \] by \(H_r^{\text{even}}\) and \(H_r^{\text{odd}}\) denote the subspaces of matrices respectively even and odd in \(t\). Let \(A\in H_r^{\text{odd}}\) be a matrix with a sufficiently small \(\| A\|\). The question is whether it is possible to find a matrix \(U\) such that \(U,U^{-1}\in H_\rho^{\text{even}}\), \(\rho<r\), and the transformation \(X=UY\) reduces the given system into \(\dot Y=0\). The paper presents the following new results: The exact applicability limit for the Kolmogorov method is found. The smoothing method is interpreted in terms of the spaces \(H_r\).