[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