Bel Hadj Fredj, Olfa; Burgos, M.; Oudghiri, M. Ascent spectrum and essential ascent spectrum. (English) Zbl 1160.47007 Stud. Math. 187, No. 1, 59-73 (2008). 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 \(a_e(T)<\infty\). Reviewer: Johannes F. Brasche (Clausthal) Cited in 3 ReviewsCited in 23 Documents MSC: 47A53 (Semi-) Fredholm operators; index theories 47A55 Perturbation theory of linear operators 47A10 Spectrum, resolvent 47A11 Local spectral properties of linear operators Keywords:spectrum; ascent; essential ascent; perturbation; semi-Fredholm PDFBibTeX XMLCite \textit{O. Bel Hadj Fredj} et al., Stud. Math. 187, No. 1, 59--73 (2008; Zbl 1160.47007) Full Text: DOI