A Note on Riesz Bases of Eigenvectors of Certain Holomorphic Operator-Functions

Operator-valued functions of the form AŽ . A QŽ . with QŽ .ŽA . 1 compact-valued and holomorphic on certain domains are considered in separable Hilbert space. Assuming that the resolvent of A is compact, its eigenvalues are simple and the corresponding eigenvectors form a Riesz basis for H of finite defect, it is shown that under certain growth conditions on QŽ .ŽA . 1 the eigenvectors of A corresponding to a part of its spectrum also form a Riesz basis of finite defect. Applications are given to operator-valued functions of the form AŽ . A BŽ D. 1Cand to spectral problems in L2Ž0, 1. of the form f Žx. pŽx, .f Žx. qŽx, .fŽx. fŽx. with, for example, Dirichlet boundary conditions.