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A characterization of dichotomy in terms of boundedness of solutions for some Cauchy problems. (English) Zbl 1168.47034
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
\[ \dot{y}(t)= -Ay(t) + e^{i\mu t}(I-P)b, \quad t\geq 0, \quad y(0) = 0. \]

47D06 One-parameter semigroups and linear evolution equations
35B35 Stability in context of PDEs
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