The Kato-type spectrum and local spectral theory. (English) Zbl 1174.47001
Summary: 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 J.Eschmeier [Proc.Edinb.Math.Soc., II.Ser.43, No.3, 511–528 (2000; Zbl 0980.47004)] 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 J.Eschmeier [loc.cit.]and M.Putinar [Integral Equations Oper.Theory 15, No.6, 1047–1052 (1992; Zbl 0773.47011)] and the theory of coherent analytic sheaves.
MSC:
| 47A11 | Local spectral properties of linear operators |
| 47A53 | (Semi-) Fredholm operators; index theories |
Keywords:
decomposable operator; semi-Fredholm operator; semi-regular operator; Kato decomposition; Bishop’s property (\(\beta \)); property (\(\delta \))References:
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