On counting the number of eigenvalues in the right half-plane for spectral problems connected with hyperbolic systems. I: Solvability of the Lyapunov equation (Q1358052)

The author studies methods of counting the number of eigenvalues in the right half-plane for the spectral problem associated with a system of differential equations which is Petrovskij hyperbolic in the strict sense. The given problem is represented as a Cauchy problem in \(L_2(0,1)\times \mathbb{R}_+\) of the form \(u_t= Au\), \(u(0)= u_0\), where the spatial operator \(A\) is closed in \(L_2(0, 1)\). The aim of this note is to investigate whether or not the eigenvalues of \(A\) lie strictly in the left half-plane and if not then how many are in the right half-plane. The notion of the index of an operator is introduced and used to characterize the stability of \(A\). The question of stability of \(A\) can be reduced to that of a selfadjoint operator \(J\) such that (1) \(\text{Re}(Au, Ju)>0\), \(u\neq 0\), the operator \(J\) is assumed to satisfy (2) \(A^*J+ JA= V\), where \(V\) is a positive definite operator. The possibility of the coincidence of the stability indices of \(A\) and \(J\) satisfying (1) and the solvability of (2) are examined on the basis of the assumption that \(A\) has finitely many eigenvalues in the right half-plane and, moreover, that the distance of the spectrum of \(A\) from the imaginary axis is greater than zero.