Basis property of a part of the system of eigenvectors of a holomorphic operator-function (Q1177803)

Let us consider the operator-function \(L(\lambda)=\lambda E+\sum_{k=0}^ \infty \lambda^ k A_ k\), where \(A_ k=A_ k^*\), \(k=1,2,\dots\) and \(A_ k\in\sigma_{p_ k}\), \(0<p_ k\leq\infty\), \(k\leq m\). If the operators \(A_ k\) (\(k>m\)) are bounded, \(a<0<b\), \(L'(\lambda)\gg 0\), \(\lambda\in[a,b]\), \(L(a)\ll 0\), \(L(b)\gg 0\), then the eigenvectors, which correspond to eigenvalues from \([a,b]\), form a basis.