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Ascent spectrum and essential ascent spectrum. (English) Zbl 1160.47007

Let \(X\) be an infinite-dimensional complex Banach space and let \({\mathcal L}(X)\) denote the algebra of all operators on \(X\). For \(T\in{\mathcal L}(X)\), let \[ a(T):=\text{ inf}\{n\geq 0:N(T^n)=N(T^{n+1})\}, \]
where the infimum over the empty set is taken to be infinite and \(N(T)\) denotes the kernel of \(T\),
\[ a_e(T):=\inf\{n\geq 0:\dim N(T^{n+1})/N(T^n)<\infty\}, \]
and
\[ E(T):=\{\lambda\in G:\text{ the resolvent of }T\text{ has a pole at }\lambda\}. \]
It is, in particular, shown that the following assertions are equivalent:
(1) \(\dim X<\infty\),
(2) \(a(T)<\infty\) for every \(T\in{\mathcal L}(X)\),
(3) \(a_e(T)<\infty\) for every \(T\in {\mathcal L}(X)\).
Let
\[ \rho_{\text{asc}}(T):=\{\lambda\in G:a(T-\lambda)<\infty\text{ and }R(T^{a(T-\lambda)+1})\text{ is closed}\}, \]
where \(R(T)\) denotes the range of \(T\),
\[ \rho_{\text{asc}}^e(T):=\{\lambda\in G:a_e(T-\lambda)<\infty\text{ and }R(T^{a_e(T-\lambda)+1})\text{ is closed}\}, \]
\[ \sigma_{\text{asc}}(T):=G\setminus\rho_{\text{asc}}(T),\quad\sigma^e_{\text{asc}}(T)=G\setminus\rho^e_{\text{asc}}(T). \]
It is also shown that the following assertions are equivalent:
(1) \(\sigma(T)\), the spectrum of \(T\), is at most countable,
(2) \(\sigma(T)\) is at most countable,
(3) \(\Sigma^e_{\text{asc}}(T)\) is at most countable. In this case, \(\sigma^e_{\text{asc}}(T)=\sigma_{\text{asc}}(T)\) and \(\sigma(T)=\sigma_{\text{asc}}(T)\cup E(T)\).
For \(R\in{\mathcal L}(X)\), it is also shown that the following assertions are equivalent:
(1) there exists a positive integer \(n\) such that \(F^n\) has finite rank,
(2) \(\sigma^{(e)}_{\text{asc}}(T+F)=\sigma^e_{\text{asc}}(T)\) for all \(T\in{\mathcal L}(X)\) commuting with \(F\),
(3) \(\sigma_{\text{asc}}(T+F)=\sigma_{\text{asc}}(T)\) for all \(T\in{\mathcal L}(X)\) commuting with \(F\).
Finally, the quasi-nilpotent part, the analytic core and the single-valued extension property are analyzed for operators \(T\) with \(a_e(T)<\infty\).

MSC:

47A53 (Semi-) Fredholm operators; index theories
47A55 Perturbation theory of linear operators
47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
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