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Entropy numbers of r-nuclear operators between \(L_ p\) spaces. (English) Zbl 0563.47013
This paper is another contribution to the study of the distribution of eigenvalues, entropy numbers, approximation numbers, etc., of particular types of operators on Banach spaces, all of which generalize classical results on summability of eigenvalues and s-numbers of various types of compact operators on Hilbert space. The main result here is that the sequence of entropy numbers of an r-nuclear operator from \(L_ p\) to \(L_ q\) \((0<r<1,1<p,q<+\infty)\) is in the Lorentz space \(\ell_{s,r}\) where \(\frac{1}{s}=\frac{1}{r}+\min \left( \begin{matrix} 1\\ 2\end{matrix} ;\frac{1}{p}\right)-\max (\frac{1}{2};\frac{1}{q})\).
Reviewer: J.R.Holub

MSC:
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47L10 Algebras of operators on Banach spaces and other topological linear spaces
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