[Author]
O. I. Reinov
[Title]
Approximation properties $AP_s$ and $p$-nuclear operators
(the case $1\le s\le \infty$)
[AMS Subj-class]
47B10 Hilbert--Schmidt operators, trace class operators,
nuclear operators, p-summing operators, etc.
[Abstract]
Banach spaces with (or without) the approximation properties
$AP_s$, $1\le s\le \infty$, are studied, --- in connection
with a question under which conditions on Banach spaces $X$
and $Y$ each operator $T: X\to Y$ with $p$-nuclear second
adjoint is $p$-nuclear itself. We give some sufficient
conditions for the positive answer to this question.
In a sense these conditions are necessary and the corresponding
counterexamples are obtained in the strongest form. In particular,
it is shown that there is a separable Banach space $V$ with
a basis such that for every $p\in [1,\infty]$ one can find
a non-$p$-nuclear operator in $V$ with $p$-nuclear second
adjoint. In previous examples of such a kind corresponding
spaces did not have even the approximation property.
[Comments]
AmSTeX, 22 pp., Russian
[Contact e-mail]
orein@orein.usr.pu.ru