A class of nonlinear equations in Hilbert space and its application to completeness problems

Abstract

The article deals with continuously differentiable operator functions \(L: [a,b] \to S(H)\) taking values in the space of selfadjoint bounded operators in a Hilbert space \(H\). It is supposed that (I) \(H\) has a decomposition \(H = H_0 \sqcup H_\emptyset\) into disjoint cones \(H_0\), \(0 \in H_0\), and \(H_\emptyset\) such that for all \(0 \neq x \in H_0\), the function \((L(\lambda)x,x)\) has a simple zero \(p(x)\) in \([a,b]\) and \((L'(p(x))x,x) > 0\); (II) \((L(\lambda)x,x) > 0\) for all \(x \in H_\emptyset\) and \(\lambda \in [a,b]\). The author states that, under these conditions, each \(\lambda_0 \in \sigma(L) \setminus \pi(L)\) is an isolated eigenvalue of finite multiplicity; moreover, \(\lambda_0\) is a simple pole of \(R(\lambda)\): \(R(\lambda) = (\lambda - \lambda_0)^{-1}P(\lambda_0) A(\lambda) + B(\lambda)\) (\(P(\lambda_0)\) is the projection on \(\text{ Ker} \, (L(\lambda_0))\)) with \(A(\lambda)\) and \(B(\lambda)\) continuous in a neighborhood of \(\lambda_0\). Furthermore, the solvability properties of the equation \(Tx = y\) with the nonlinear operator \(Tx = L(p(x))x\), \(x \in H_0 \setminus\{0\}\)), \(T0 = 0\), are studied. At the end of the article, two examples with \(H = {\mathbb R}^2\) are presented.