Joint spectrum of commuting self-adjoint operators and tests for propriety and stability for differential-operator equations (Q1174887)

Some qualitative properties for the equation \[ y''+Ay'+By=0, \qquad t\in[0,\infty), \] where \(A\), \(B\) are commuting selfadjoint operators in a separable Hilbert space, are established; these properties are the correctness of the Cauchy problem, the stability, the asymptotic and exponential stability. For this purpose some criteria which assure that the operator \(B\) depends, in a defined sens, on \(A\), using the joint spectrum of the operators \(A\), \(B\), are proved. The results concerning the correctness and the stability are applied in the case of any partial differential equations.