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Lower s-numbers and their asymptotic behaviour. (English) Zbl 0695.47016

Let T be a bounded linear operator on a Banach space X, with unit ball U, and let \[ m(T)=\inf \{\| Tx\|:\quad \| x\| =1\},\quad q(T)=\sup \{\epsilon \geq 0:\quad TU\supset \epsilon U\}. \] Further, for \(r=1,2,..\). let \[ m_ r(T)=\sup \{m(T+F):\quad rank F<r\}, \]
\[ q_ r(T)=\sup \{q(T+F):\quad rank F<r\}, \]
\[ g_ r(T)=\max \{m_ r(T),q_ r(T)\}. \] The author studies the asymptotic behaviour of \(g_ r(T)\). The main result is as follows:
If \(\omega_ r(T)\) is the r-th jumping point of the minimum index function of a semi-Fredholm operator T, then \[ | \omega_ r(T)| =\lim_{k}g_{kn+r}(T^ k)^{1/k}, \] where n is the stability index of T.
Reviewer: U.Schlotterbeck

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A53 (Semi-) Fredholm operators; index theories
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