Norm estimate for the inverse of a linear pencil with Hilbert–Schmidt operators

Abstract

Let H be a Hilbert space with the unit operator I . The paper deals with the pencil T (z) = A + z B (z ∈ C), where A and B are Hilbert-Schmidt operators in H . We suggest a norm estimate for the inverse of I − T (z). By that estimate, bounds for the spectra of I − T (.) are established and perturbations by bounded operators are investigated. Applications to boundary value problems are also discussed.