Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On linear operators with s-nuclear adjoints: $0< s \le 1$   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.

12
The categories of this paper are: nucl-th
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph