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The asymptotic behaviour of the reduced minimum modulus of a Fredholm operator

Authors: K.-H. Förster and M. A. Kaashoek
Journal: Proc. Amer. Math. Soc. 49 (1975), 123-131
MSC: Primary 47A55; Secondary 47B30
MathSciNet review: 0372660
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Abstract: Let $ \gamma (S)$ denote the reduced minimum modulus of a linear operator $ S$ acting in a complex Banach space $ X$, and let $ I$ denote the identity on $ X$. In this paper it is shown that for a (not necessarily bounded) Fredholm operator $ T$ acting in $ X$, the limit $ {\lim _{n \to \infty }}\gamma {({T^n})^{1/n}}$ exists and is equal to the supremum of all positive numbers $ \delta $ such that the dimension of the null space and the codimension of the range of $ T - \lambda I$ are constant on $ 0 < \vert\lambda \vert < \delta $.

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Keywords: Fredholm and semi-Fredholm operators, reduced minimum modulus, stability theory, spectrum
Article copyright: © Copyright 1975 American Mathematical Society