On linear operators with s-nuclear adjoints: $0<s\le 1$

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 $X^{\ast }$ or $Y^{\ast \ast \ast }$ has the approximation property of order $s,$ $A{P}_s,$ 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 $T:Z_0^{\ast \ast }\to Z_0$ so that $Z_0^{\ast \ast }$ has a Schauder basis, $Z_0^{\ast \ast \ast }$ has the $A{P}_s$ for every $s\in (0,r)$ and $T^{\ast }$ is $r$-nuclear.

11 pages, AMS TeX

References in corpus (1)

• An example of an asymptotically Hilbertian space which fails the approximation property