巴拿赫空间上线性算子特征值个数的估计
Estimating the number of eigenvalues of linear operators on Banach spaces
摘要：
设$L_0$是Banach空间上的有界算子，考虑扰动$L=L_0+K$，其中$K$是紧的.本文研究了扰动算子$L$的本质谱的补集上的特征值个数的界.我们的结果可以看作是紧算子特征值分布的经典结果的广泛推广，它对应于$L_0=0$情形.它们也推广了以前关于Hilbert空间中算子的结果。我们的方法采用复杂的分析和一个新的有限维减少，使我们能够避免使用现有的理论在巴拿赫空间，这将需要强有力的限制$K$的行列式。提出了几个悬而未决的问题，我们的结果的锐度，并构造了一个例子，表明有一些本质的区别，在一般的巴拿赫空间的运营商的特征值的可能分布相比，希尔伯特空间的情况。
英文摘要：
Let $L_0$ be a bounded operator on a Banach space, and consider a perturbation $L=L_0+K$, where $K$ is compact. This work is concerned with obtaining bounds on the number of eigenvalues of $L$ in subsets of the complement of the essential spectrum of $L_0$, in terms of the approximation numbers of the perturbing operator $K$. Our results can be considered as wide generalizations of classical results on the distribution of eigenvalues of compact operators, which correspond to the case $L_0=0$. They also extend previous results on operators in Hilbert space. Our method employs complex analysis and a new finite-dimensional reduction, allowing us to avoid using the existing theory of determinants in Banach spaces, which would require strong restrictions on $K$. Several open questions regarding the sharpness of our results are raised, and an example is constructed showing that there are some essential differences in the possible distribution of eigenvalues of operators in general Banach spaces, compared to the Hilbert space case.
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