Compact Perturbations of a Strongly Positive Operator

Abstract

Let X be a real Hilbert space and $$X^{*}$$ its dual space. Let A be a strongly positive operator from X to $$X^{*}$$ . Sufficient conditions are provided to assert that a compact perturbation of A reaches a fixed value at least once and at most finitely many times. When the compact perturbation is linear, the value is reached just once. The same conclusions are obtained when operators map the space X into itself. As an application of our results two examples are given related to some types of integral equations. The proof of results is constructive and is based upon a continuation method.