Geodesic
The Kato-type spectrum and local spectral theory
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 831-842.

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Let $T\in {\mathcal{L}}(X)$ be a bounded operator on a complex Banach space $X$. If $V$ is an open subset of the complex plane such that $\lambda -T$ is of Kato-type for each $\lambda \in V$, then the induced mapping $f(z)\mapsto (z-T)f(z)$ has closed range in the Fréchet space of analytic $X$-valued functions on $V$. Since semi-Fredholm operators are of Kato-type, this generalizes a result of Eschmeier on Fredholm operators and leads to a sharper estimate of Nagy’s spectral residuum of $T$. Our proof is elementary; in particular, we avoid the sheaf model of Eschmeier and Putinar and the theory of coherent analytic sheaves.
Classification : 47A11, 47A53
Keywords: decomposable operator; semi-Fredholm operator; semi-regular operator; Kato decomposition; Bishop’s property ($\beta $); property ($\delta $) 
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     title = {The {Kato-type} spectrum and local spectral theory},
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Miller, T. L.; Miller, V. G.; Neumann, M. M. The Kato-type spectrum and local spectral theory. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 831-842. http://geodesic.mathdoc.fr/item/CMJ_2007__57_3_a3/