Spectral theory for linear non-autonomous differential equations
The linear non-autonomous equations \(x' = A(t)x\) having bounded growth and decay are considered \((A(t)\) is a matrix function). The spectral mapping theorem \(\sigma (T^ h) = \exp h \sigma' (L)\) and rotational invariance are proved by introducing a semigroup of the operators \(T^ h\) using the relations \((T^ hv)(t) = X(t,t-h) v(t-h)\) and \(\lim v(t) = 0\) as \(t \to \infty\). As examples the difference equation \(x(t) = Bx(t-1) + f(t)\), the linearly perturbed equation \(dy/dt = (A(t) + B(t))y\), the equation \(dx/dt = Ax\) where \(A\) is a matrix and the Sacker-Sell spectrum are discussed.
scientific article; zbMATH DE number 703331