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On the compact non-nuclear operator problem. (English) Zbl 0687.47012

The problem reads: Let X, Y be two infinite dimensional Banach spaces.:
Does there always exist a compact non-nuclear operator f: \(X\to Y?\)
This question is settled in the negative by showing that any compact operator from Pisier’s space P into its dual \(P^*\) is nuclear. The result uses standard properties of Pisier’s space and an interesting approximation result.
Reviewer: K.John

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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