This is an announcement for the paper "On linear operators with ${\ssize\bold s}$-nuclear adjoints: $0<{\ssize s}\le 1$" by O.I. Reinov.
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.
Archive classification: math.FA
Remarks: 11 pages, AMS TeX
Submitted from: orein51@mail.ru
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1311.2270
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