Instability spectrum of an operator bundle (Q1814574)

The spectrum of an operator function of the form \[ L(\lambda)=\lambda^ 2 I+\lambda(\alpha A+R)+A+C-\hat\alpha(\lambda)A. \] is investigated. Here \(A\) is an (unbounded) selfadjoint operator in a Hilbert space, operators \(RA^{-1}\) and \(CA^{-1}\) are compact, \(R\) is accretive and \(C\) is symmetric. The function \(\hat a(\lambda)\) is the Laplace transform of a real valued function \(a(t)\in L^ 1(0,\infty)\) such that \(ta(t)\in L^ 1(0,\infty)\). Additionally, it is assumed that \(1-\hat\alpha(0)>0\), \(\alpha\omega-\hbox{Im}\hat\alpha(i\omega)>0\) \((\omega>0)\) and \(\alpha- a'(0)>0\). The author gives a description of the part of the spectrum contained in the right half-plane.