Linear pencils and quadratic programming problems with a quadratic constraint

Abstract

Given bounded selfadjoint operators A and B acting on a Hilbert space, consider the linear pencil P(λ)=A+λB, λ in R. The set of parameters λ such that P(λ) is a positive (semi)definite operator is characterized. These results are applied to solving a quadratic programming problem with an equality quadratic constraint (or a QP1EQC problem).