Singular numbers of a boundary-value problem on the halfline for a linear system of ordinary differential equations
The singular numbers of a closed linear operator in a Hilbert space with dense domain of definition are defined and their basic properties are derived. Operators connected with the boundary value problems for the systems of differential equations are investigated as well as the singular spectrum. Operators \(T_ 0\in L_ 2(R)\) defined as \(T_ 0y(t):=dy(t)/dt-Ay(t),\) where A denotes an \(n\times n\) matrix without imaginary eigenvalues, and T which complies in addition with the boundary condition \(Ly(0)=0\) are studied in more details. It is proved e.g. that under certain conditions the singular spectrum of \(T_ 0\) or T, respectively, is formed by the halfline \([\sigma_*,\infty)\) or \([\sigma_*,\infty)\) and finite number points, respectively, where \(\sigma_*=\min_{\omega}\sigma_ 1(i\omega I-A),\) \(\delta_ 1(i\omega I-A)\) being the least singular number of the matrix \(i\omega\) I-A.
scientific article; zbMATH DE number 4142486