Zada, Akbar A characterization of dichotomy in terms of boundedness of solutions for some Cauchy problems. (English) Zbl 1168.47034 Electron. J. Differ. Equ. 2008, Paper No. 94, 5 p. (2008). Summary: We prove that a quadratic matrix of order \(n\) having complex entries is dichotomic (i.e., its spectrum does not intersect the imaginary axis) if and only if there exists a projection \(P\) on \(\mathbb{C}^n\) such that \(Pe^{tA}=e^{tA}P\) for all \(t\geq 0\) and, for each real number \(\mu\) and each vector \(b\in\mathbb{C}^n\), the solutions of the following two Cauchy problems are bounded: \[ \dot x(t)=Ax(t)+e^{i \mu t}Pb,\quad t\geq 0, \quad x(0) = 0 , \] and \[ \dot{y}(t)= -Ay(t) + e^{i\mu t}(I-P)b, \quad t\geq 0, \quad y(0) = 0. \] Cited in 9 Documents MSC: 47D06 One-parameter semigroups and linear evolution equations 35B35 Stability in context of PDEs Keywords:stable and dichotomic matrices; Cauchy problem; spectral decomposition theorem PDFBibTeX XMLCite \textit{A. Zada}, Electron. J. Differ. Equ. 2008, Paper No. 94, 5 p. (2008; Zbl 1168.47034) Full Text: EuDML EMIS