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On linear operators with s-nuclear adjoints: $0< s \le 1$ (1311.2270v1)

Published 10 Nov 2013 in math.FA

Abstract: If $ s\in (0,1]$ and $ T$ is a linear operator with $ s$-nuclear adjoint from a Banach space $ X$ to a Banach space $ Y$ and if one of the spaces $ X*$ or $ Y{***}$ has the approximation property of order $s,$ $AP_s,$ then the operator $ T$ is nuclear. The result is in a sense exact. For example, it is shown that for each $r\in (2/3, 1]$ there exist a Banach space $Z_0$ and a non-nuclear operator $ T: Z_0{**}\to Z_0$ so that $Z_0{**}$ has a Schauder basis, $ Z_0{***}$ has the $AP_s$ for every $s\in (0,r)$ and $T*$ is $r$-nuclear.

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