Let \(\mathcal {W}\) be an ideal of compact operators A in a Banach space X satisfying the condition \(N_\mathcal {W}(A)=\sum _{k=1}^{\infty }x_k(A)<\infty \), where \(x_k(A)\)\((k=1, 2, \ldots )\) are the Weyl numbers of A. It is proved that for all \(A\in \mathcal {W}\) and any regular \(\lambda \ne 0\) of A, the inequality $$\begin{aligned} \Vert \det \;(I-\lambda ^{-1}A)(\lambda I-A)^{-1}\Vert \le \frac{ c }{ |\lambda | } \exp \;\left[ \frac{ 2cN_\mathcal {W}(A) }{ |\lambda | }\right] \end{aligned}$$ is valid, where \(c=\sqrt{2e}\). Applications of this inequality to spectrum perturbations are also discussed.

中文翻译：

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Banach空间中算子的求解器和行列式之间的不等式

令 \(\mathcal {W}\) 是 Banach 空间 X 中紧致算子 A 的理想满足条件 \(N_\mathcal {W}(A)=\sum _{k=1}^{\infty } x_k(A)<\infty \)，其中 \(x_k(A)\)\((k=1, 2, \ldots )\) 是 A 的外尔数。 证明对于所有 \(A\在 \mathcal {W}\) 和 A 的任何正则 \(\lambda \ne 0\) 中，不等式 $$\begin{aligned} \Vert \det \;(I-\lambda ^{-1}A) (\lambda IA)^{-1}\Vert \le \frac{ c }{ |\lambda | } \exp \;\left[ \frac{ 2cN_\mathcal {W}(A) }{ |\lambda | }\right] \end{aligned}$$ 有效，其中 \(c=\sqrt{2e}\)。还讨论了这种不等式在频谱扰动中的应用。