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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
$\dot{y}(t)= -Ay(t) + e^{i\mu t}(I-P)b, \quad t\geq 0, \quad y(0) = 0.$

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