A note on Riesz bases of eigenvectors of certain holomorphic operator-functions

Operator-valued functions of the form A(lambda) := A - lambda + Q(lambda) with lambda --> Q(lambda)(A - mu)(-1) compact-valued and holomorphic on certain domains Omega subset of C 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 K of finite defect, it is shown that under certain growth conditions on parallel toQ(lambda)(A - lambda)(-1)parallel to the eigenvectors of A corresponding to a Dart of its spectrum also form a Riesz basis of finite defect. Applications are given to operator-valued functions of the form A(lambda) = A - lambda + B(lambda - D)C-1 and to spectral problems in L-2(0, 1) of the form -f " (x) + p(x, lambda )f'(x) + q(x, lambda )f(x.) = lambdaf(x) with, for example, Dirichlet boundary conditions. (C) 2001 Academic Press.