Second-order linear differential equations in a Banach space and splitting operators
Let \({\mathcal X}\) be a Banach space, let \({\mathcal X}^2={\mathcal X}\times{\mathcal X}\). In the space \(C_b(\mathbb{R},{\mathcal X})\) consider the second-order linear differential operator \[ Lx=\ddot x+ B_1\dot x+ B_2x \] with the domain \(D(L)= C^{(2)}_b(\mathbb{R},{\mathcal X})\). Here, \(B_1\), \(B_2\) are bounded operators in \({\mathcal X}\). The operator \(L\) can be represented as the first-order differential operator in the space \(C_b(\mathbb{R},{\mathcal X}^2)\), \[ \mathbb{L} y=\dot y+\mathbb{B} y,\quad \mathbb{B}= \begin{pmatrix} 0 & -I\\ B_2 & B_1\end{pmatrix} \] with the domain \(D(\mathbb{L})= C^{(2)}_b(\mathbb{R},{\mathcal X}^2)\). Consider the related quadratic operator equation \[ X^2+ B_1X+ B_2= 0. \] Assume that the quadratic equation has two separated roots \(\Lambda_1\), \(\Lambda_2\), i.e. \(\Lambda_1-\Lambda_2\) is invertible. Then it is proved that the operator \(\mathbb{B}\) is similar to the operator \(\Lambda=(-\Lambda_1)\oplus(-\Lambda_2)\) and the operators \(L\), \(\mathbb{L}\) are invertible iff \((\sigma(\Lambda_1)\cup(i\mathbb{R})= \emptyset\). If \[ (\sigma(\Lambda_1)\cup\sigma(\Lambda_2))\cap (i\mathbb{R})= \{\lambda_1,\dots, i\lambda_m\} \] then there exist projection-valued uniformly continuous functions \(\mathbb{P}_k(t)\in C_b(\mathbb{R}_+,{\mathcal X})\) such that the representation \[ (x(t),\dot x(t))= \sum^m_{k=1} e^{i\lambda_kt} \mathbb{P}_k(t) (x(0),\dot x(0)),\quad t\geq 0 \] is valid for every solution \(x:\mathbb{R}_+\to{\mathcal X}\) to be homogeneous differential equation \(Lx=0\).
scientific article; zbMATH DE number 6835433
scientific article