We consider the nonlinear eigenvalue problem $$Lx + \varepsilon N(x) = \lambda Cx, \quad \|x\|=1,$$ where $\varepsilon,\lambda$ are real parameters, $L, C\colon G \to H$ are bounded linear operators between separable real Hilbert spaces, and $N\colon S \to H$ is a continuous map defined on the unit sphere of $G$. We prove a global persistence result regarding the set $\Sigma$ of the solutions $(x,\varepsilon,\lambda) \in S \times \mathbb R\times \mathbb R$ of this problem. Namely, if the operators $N$ and $C$ are compact, under suitable assumptions on a solution $p_*=(x_*,0,\lambda_*)$ of the unperturbed problem, we prove that the connected component of $\Sigma$ containing $p_*$ is either unbounded or meets a triple $p^*=(x^*,0,\lambda^*)$ with $p^* \not= p_*$. When $C$ is the identity and $G=H$ is finite dimensional, the assumptions on $(x_*,0,\lambda_*)$ mean that $x_*$ is an eigenvector of $L$ whose corresponding eigenvalue $\lambda_*$ is simple. Therefore, we extend a previous result obtained by the authors in the finite dimensional setting.

Our work is inspired by a paper of R. Chiappinelli concerning the local persistence property of the unit eigenvectors of perturbed self-adjoint operators in a real Hilbert space.

中文翻译：

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Hilbert空间中被摄动特征值问题的单位特征向量的全局持久性

我们考虑非线性特征值问题$$ Lx + \ varepsilon N（x）= \ lambda Cx，\ quad \ | x \ | = 1，$$其中$ \ varepsilon，\ lambda $是实参，$ L，C \冒号G \ to H $是可分离的实Hilbert空间之间的有界线性算子，而$ N \冒号S \ to H $是在$ G $单位球面上定义的连续映射。我们证明了关于解决方案集合\\ Sigma $的全局持久性结果$（x，\ varepsilon，\ lambda）\ in S \ times \ mathbb R \ times \ mathbb R $ 即，如果算子$ N $和$ C $是紧凑的，则在适当的假设下，在无扰动问题的解$ p _ * =（x _ *，0，\ lambda _ *）$上，我们证明了$ \的连通分量包含$ p _ * $的Sigma $是无界的或满足三元$ p ^ * =（x ^ *，0，\ lambda ^ *）$和$ p ^ * \ not = p _ * $。当$ C $是恒等式且$ G = H $是有限维时，对$（x _ *，0，\ lambda _ *）$的假设意味着$ x _ * $是$ L $的特征向量，其对应的特征值$ \ lambda _ * $很简单。因此，我们扩展了作者在有限维设置中获得的先前结果。

我们的工作受到R. Chiappinelli的一篇论文的启发，该论文涉及在实际Hilbert空间中被扰动的自伴算子的单位特征向量的局部持久性。