\input trtrtrtr.tex

\comment
\head
  \bf Approximation Properties:\, Topological Aspect. With Appendix
    to a Grothendieck Conjecture
\endhead
\med
\centerline{\bf Oleg I. Reinov}
			  \small
 \centerline{Abdus Salam School of Mathematical Sciences}
 \centerline{and}
 \centerline{St. Petersburg State University}
\bigp
	\endcomment

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%$$$$$$$$$$$$$$$$$$$$$
\CenteredTagsOnSplits                        \NoBlackBoxes



\NoRunningHeads
%\pageno=1
%\footline={\hss\tenrm\folio\hss}

%        \topmatter
%        \title {
%Approximation Properties: \, Topological Aspect
% $\text With Application to a Grothendieck Conjecture$
%}
%        \endtitle
%          \author {Oleg Reinov${ }^\dag$\
%        }
%\endauthor

   \topmatter
        \title {
Approximation Properties: \ Topological Aspect\\
  {\eightbf  (With Application
    to a Grothendieck Conjecture)}
}
        \endtitle
          \author {Oleg Reinov${ }^\dag$\
        }
\endauthor




  \comment
\abstract
For a tensor norm $\al$ on the class of all Banach spaces
is is a natural way to define the Banach operator ideal
$\N_\al$ associated with the tensor norm.
We give sufficient conditions for an operator to be in
$\N_\al$ if its second adjoint possesses this property.
We show one  way to get from this
the negative answers to some questions of A. Defant and K. Floret:
their tensor norms $g_\infty,$ $w_1$ and $w_\infty$
are not totally accessible.
\endabstract
    \endcomment

{\thanks{\enskip    ${ }^\dag$This research  was done with partial support
by the Fond RFFI Grant  06-01-00457}\endthanks}



       \endtopmatter

\address\newline
Oleg I. Reinov \newline
{}\quad Department of Mathematics\newline
St Petersburg University\newline
 St Petersburg, Russia;\newline
%St Peterhof, Bibliotech pl 2\newline
%198904  St Petersburg, Russia;\newline
{}\quad Abdus Salam School of Mathematical Sciences\newline
GC University, Lahore, Pakistan\newline
\endaddress

\email
orein51\@mail.ru\newline
\endemail


\document

\baselineskip=18pt

\footnote""{${ }^\maltese$
AMS Subject Classification 2000:
46B28.
Spaces of operators; tensor products;
approximation properties
}
\footnote""{${ }$
Key words: dually $p$-nuclear operators, approximation property,
compact approximation, nuclear tensor norms, tensor products.
}
 \bigp

%----------------------------


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\cc{\circ}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In 1932, Stefan Banach posed his famous problem
on the existence of a (Schauder)
basis in any separable Banach space
(the first example of a concrete basis in nonhilbertian space,
namely, in $C[0,1]$ was given by Schauder).
Since that time many mathematicians tried to find bases and did it
for most classical Banach spaces
(A. Haar, J. Schauder, J. Marcinkiewicz, and others).
But the general Problem was still unsolved. For a long time,
there were even no thoughts how to deal with this very important Problem.
The positive solution would yield
a lot of applications, would lead to positive solutions of most
of well known open problems
in the corresponding fields.

In 1955, there appeared the famous fundamental work of Alexander Grothendieck
"Produits tensoriels topologiques et espaces nucl\'eaires" in
Mem. Amer. Math. Soc. 16 (1955).
In this treatise, A. Grothendieck has introduced and investigated
a lot of new notions,
especially, the notions of different approximation properties
of spaces and operators
(so-called AP and BAP; they generalize the property
of a Banach space to have a basis).
One of the nice consequence: if $X$ has the AP then
every compact operator in $X$ is
in the closure of the set of all finite rank operators.

In his great work, A. Grothendieck has posed
(in the very end of the treatise)
more than 15 main unsolved Problems.
 Among them, Problems:\
 1)\ of existence of a Banach space without AP or BAP;
 2)\ of existence of a space with AP, but without BAP;
 3)\ of equivalence of AP and BAP for all spaces.
 Concerning the last question,
he had showed that for Banach spaces with weakly compact
 identity maps (for reflexive spaces) these two properties are the same.
 This led him to ask:\,
 4)\ whether any weakly compact operator with AP possesses also BAP?


Recall some definitions.
The Banach space $X$ has the {\it approximation property} $\AP$
if for each compact $K\sbs X$ and for every $\e>0$ there exists
a finite rank operator $R$ in $X$ such that $||x-Rx||\le\e$ for any
$x\in K.$ In other words, $X$ has the $\AP$ iff the identity map in $X$
can be uniformly, on each compact, approximated by finite rank operators.
$X$ has the {\it bounded approximation property} $\BAP$
if there is a constant $C>0$ so that
for each compact $K\sbs X$ and for every $\e>0$ there exists
a finite rank operator $R$ in $X$ such that $||x-Rx||\le\e$ for any
$x\in K$ and $||R||\le C.$
In his work, A. Grothendieck was able to reformulate these properties
in terms of introduced by him the so--called projective and
injective tensor products, as well as in terms of his "nuclear" and "integral"
operators.

The problems of A. Grothendieck were solved by different mathematicians,
firstly (and mainly) by P. Enflo (the famous example of a reflexive Banach space
without $\AP;$ se, e.g., [10]).

The natural generalizations of $\AP$ and $\BAP$
(the properties $\AP_p, BAP_p, AP_p^{dual}$ etc.)
were considered by
P. Saphar [21], the author [11--13], [16] and others.
All these properties were firstly defined in terms of some tensor products
(generalizations of Grothendieck ones). One of the question of P. Saphar
was to describe the $\AP_p$ in terms of something like "compact convergence"
for absolutely $p$-summing operators. It was done in [15] by the author
of this paper. The further investigations in this direction can be found in
[1], [11--13] etc.

This paper is devoted, mainly,
to the proceeding in the same direction for the class
(ideal) $\Pi_p^d$
of dually absolutely-$p$-summing operators,
and giving the connections between some notions of "compact convergence"
of type $\pi_p^{dual}$ and the properties of tensor products, with applications.
We will apply the results to the investigation of a natural question:
when a Banach space has the generalized approximation properties (such as
introduced below the properties $\AP^p)?$
We give here some sufficient and some necessary conditions for the space
to have $\AP^p$ as well as construct some (counter)examples to the
$\AP^p$-approximation problems.
In the very end of the paper, we apply our results to give a new proof
of a much more stronger result than one from my paper [14]
(concerning a conjecture of A. Grothendieck from his fundamental work [3]).
Answering in negative to the Grothendieck question whether every
weakly compact operator with the $\AP$ has also the $\BAP,$
we have constructed in [14] the example of a {\it compact}\ operator
with $\AP,$ but without $\BAP.$
Here, in Theorem 10, we show that there exist the operators of such a kind,
belonging even to the classes of dually quasi-$p$-nuclear
(hence, compact) operators.
\small

{\bf Acknowledgements.}\,
I would like to bring my sincere acknowledgements to the
Abdus Salam School of Mathematical Sciences,
and especially to Dr. Professor A.D.R. Choudary,
for providing me with excellent working conditions during my stay
in Lahore in 2007--2008.
%\small
 \bigp

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%----------------------------

\centerline{\bf \S 1. Preliminaries}

\small

We use standard notation of the theories of Banach spaces,
of linear operators and of operator ideals
(see, e.g., [10]). The Banach spaces under consideration will be denoted
usually by $X, Y, Z, E,$ etc.
Elements of these spaces are denoted by corresponding small letters,
for example, $x\in  X, y \in Y, x_0\in X_1, y_k\in Y etc.$
Banach duals to the spaces $X,Y$ etc.
are denoted by $X^*, Y^*$ etc.
 The value of a functional $f \in X^*$ or of $x'\in X^*$ at an element
$x\in X$ is denoted
by $\<f,x\>$  or $f (x);$ by $\<x',x\>$ or $\<x,x'\>$
(everywhere, we consider the space $X$ as a subspace of its
second dual $X^{**}$ without introducing a special notation
for the canonical isometric
injection $X\to X^{**}).$
$\operatorname{L}(X,Y)$ is
the space of all (linear continuous) operators from $X$ to $Y$
with the usual operator norm.
If $A\sbs X,$ then $\Gamma (A)$ is a absolutely convex hull
of $A,$ $\ove A$ is the  closure of the set
$A$ in $X.$ If we need to show that the closure is taken
in a topology $\tau$ on $X,$ we write
${\ove A}^{\tau}.$
For an absolutely convex closed bounded set $K\sbs X,$
we denote by $X_K$ the Banach space $\cup_{n=1}^\infty nK$
with he unit ball $K;$
$\Phi_K: X_K\to X$ is then the canonical injection.
The linear space of all finite rank operators from $X$ to $Y$
is considered as a tensor product
$X^*\ot Y.$
Finely, if needed,
the norm of an $x\in X$ in the space $X$ is denoted also by $||x||_X.$

Recall that every Banach space $X$ is isometric to a factor-space of a space
$ l_1(\Gamma).$
% (one cat take as a set $ \Gamma$ the closed unit ball
%$B_X;$ then the operator
%$ Q(\alpha_x):=\sum\limits_{x\in B_X} \alpha_x x$ is a factor-map
%from $ l_1(B_X)$ onto $ X$).
%Every Banach space is isometric to a subspace of a
%$C(K)$ of of a $ l_{\infty}(\Gamma)$,
%for example, $ l_{\infty}(B_X)$.
Recall also that an operator $ T\in\operatorname{L}(X,Y)$ is weakly compact
iff $ T(X^{**})\subset Y$.
%By definition, $T$ is (weakly) compact if it takes bounded subsets
%to (weakly) relatively subsets.

%Notations $l_p, c_0 $ are also standard; the (quasi)norms in these spaces
%(as well as in the spaces of type $ L_p(\mu)$) are denoted by
%$ \|\alpha\|_p=\|\alpha\|_{l_p}$ for $ \alpha=(\alpha_k)\in l_p$,
%$ p\in[1,\infty]$, and
%$ \|\alpha\|_\infty=\|\alpha\|_{c_0}$ in the case of the space $ c_0.$

If $ X$ is a Banach space and
$ \mu$ is a measure then by
$ L_p(\mu; X)$, $ 0< p\leqslant \infty$ we understand
the $ L_p$-space of all (equivalent classes of) strongly
$ \mu$-measurable p-summable functions.
In the case where the measures are discrete,
we use also the notations of type
$ l_p(X)$, $ l_p(\Gamma; X)$, $ c_0(X)$, etc.
For the quasinorm of  a sequence  $ (x_k)_k$ from
the space $ l_p(X)$, we use the notation
$ \alpha_p(x_k):= \left(\sum\limits_k \|x_k\|^p\right)^{1/p}$
(if $p=\infty,$ the corresponding changes are needed).
Recall that
$ l_p(\Gamma;X)^*=l_{p'}(\Gamma;X^*)$ and $ c_0(\Gamma;X)^*=l_1(\Gamma;X)$
for all finite $ p\geqslant 1$.
A family $(x_k)_{k=1}^\infty\subset X$,
for which the value $ \alpha_p(x_k)$ is finite,
is said to be {\it absolutely $ p$-summable}.
A family $ (x_k)_{k=1}^\infty$ is called
 {\it weakly $ p$-summable,}\
if  $ (\langle x_k, x' \rangle)\in l_p$ for all $ x'\in X^*$.
We set
$$
\varepsilon_p(x_j):=
\sup_{\|x'\|\leqslant 1} \biggl(\sum\limits_j
|\langle x_j,
x'\rangle |^p\biggr)^{1/p}.
$$
This is a quasinorm (a norm if  $ p\geqslant 1$) in the space
$ l_p \{ X\}$ of all weakly $ p$-summable sequences
in $ X$.

We use the following notations for the classical operator ideals
(all the information on the theory of operator ideals can be found in
[10]; in our work, we follow, however, the other notation and terminology;
see [8, 9]):

$[\operatorname{\Pi}_p,\pi_p]$ ---
the ideal of the absolutely $ p$-summing operators;

$[\operatorname{QN}_p,\pi_p]$ ---
the ideal of quasi-$p$-nuclear operators;

$[\operatorname{N}_p, \nu_p]$ ---
the ideal of $p$-nuclear operators;

$[\operatorname{ I}_p,i_p]$ ---
the ideal of (strictly) $ p$-integral operators;

$[\operatorname{\Pi}_p^d,\pi_p^d]$ ---
the ideal of the dually absolutely $ p$-summing operators;

$[\operatorname{QN}_p^d,\pi_p^d]$ ---
the ideal of dually quasi-$p$-nuclear operators;

$[\operatorname{N}^p, \nu^p]$ ---
the ideal of dually $p$-nuclear operators;

$[\operatorname{ I}^p,i^p]$ ---
the ideal of (strictly) dually $ p$-integral operators.
\small

Recall some important definitions (see [8--10,17--18]).
Let $ T\in \operatorname{L}(X,Y)$.
For $ 0< p\leqslant \infty,$
the operator  $ T:X\to Y$ is called
{\it absolutely $p$-summing},  if there is a constant $C>0$
such that, for any finite family
$\{x_n\}^M_{n=1} \subset X,$ the following inequality holds
$ \alpha_p(Tx_n) \leqslant C
\varepsilon_p(x_n);$ corresponding norm
($ \inf C$) is denoted by $ \pi_p(T)$.
Note that $ [\Pi_{\infty}, \pi_{\infty}]$
is exactly the operator ideal $ [\operatorname{L},\|\cdot\|]$.
%%%============
For $ 0< p\leqslant \infty,$
the operator  $ T:X\to Y$ is called
 {\it quasi-$p$-nuclear},
$ T\in \operatorname{ QN}_p(X,Y)$, if for some isometric embedding
 $ i:Y\to L_\infty(\mu)$ the composition
$iT$ is in  $ \operatorname{N}_p(X, L_\infty(\nu))$.
The quasinorm (the norm if $ p\geqslant1$) in $\operatorname{ QN}{_p}(X,Y)$
is induced from the space
$ \operatorname{\Pi}_p(X, L_\infty(\nu))$.

%%%========
Generally, let $ \operatorname{ A}$ be an operator ideal with a norm
$a.$
By $ \operatorname{ A}^{\operatorname{dual}},$ we denote the dual
(to $ \operatorname{ A})$ ideal:
$ T\in \operatorname{ A}^{\operatorname{dual}} $
iff
$T^*\in \operatorname{ A}$.
The ideal $ \operatorname{ A}^{\operatorname{dual}}$
with the norm
$a^{\operatorname{dual}}(T):=a(T^*)$
is a normed operator ideal.
Let us denote by $ a^*$ the norm in the conjugate ideal
$ [\operatorname{ A},a]^*=[ \operatorname{ A}^*,a^*]$
[10, p. 9.1] .
For example, $\Pi_p^d=\Pi_p^{\operatorname{dual}},$
$\QN_p^d=\QN_p^{\operatorname{dual}},$
$[\Pi_p^d,\pi_p^d]^*=[\I^{p'}, i^{p'}].$
Here and everywhere, $1/p+1/p'=1.$
%%===================
Consider some important examples.

Let $ p\in[1,\infty]$.
We say that an operator
$ T\in \operatorname{L}(X,Y)$ {\it belongs to the ideal}
$ \operatorname{N}^p$, if it can be represented in the form
$$
T:= \sum\limits_{n=1}^{\infty} x'_n\otimes y_n,
$$
where the sequences $ (x'_n)$ and $ (y_n)$ are such that
$ \varepsilon_{p'}(x'_n)<\infty$, $ \alpha_p(y_n)<\infty$.
With the norm $ \nu^p(T):= \inf \varepsilon_{p'}(x'_n)\alpha_p(y_n)$
the class $ \operatorname{N}^p$ is a normed operator ideal.
This is almost dual ideal to the ideal $ \operatorname{N}_p;$
the ideal, which is dual to $ \operatorname{N}_p,$ is exactly
the ideal $ (\operatorname{N}^{p})^{ \operatorname{ reg}}$
(see [10] for notation),
and $ T\in \operatorname{N}_p^{\operatorname{dual}}(X,Y)$ iff
$ T\in \operatorname{N}^p(X,Y^{**})$.
Let us give other characteristics of the operators $T$ from
$ \operatorname{N}^p(X,Y)$.
An operator $ T$ is an  $ \operatorname{N}^p$-operator iff
it factors in the following way:
$$
\CD
X @>A>> l_{p'} @>{\Delta}>> l_1 @>B>> Y,
\endCD
$$

\noindent
where $ \Delta$ is a  diagonal operator with a diagonal  from $ l_{p}$,
$ A$ and $B$ are the operators of norm 1; moreover,
$ \nu^p(T)$ is just the norm of the diagonal in $ l_p$.

Let us say that an
operator $ T\in \operatorname{L}(X,Y)$
{\it belongs to the ideal}
$ \operatorname{I}^p$, if it admits a factorization of the kind
$$
\CD
X @>A>> L_{p'}(\mu) \overset{j}\to{\hookrightarrow} L_1(\mu)
@>B>> Y,
\endCD
$$
where $ \mu$ is a probability measure, $ j$ is the identity injection,
$ A$ and $ B$ are continuous operators.
We put
$ i^p(T)= \inf \|A\|
\|B\|$,
where the $ \inf$ is taken over all the factorizations of  $ T$
of the mentioned kind.
%%%===================
\small

Now, finally, some important words on the notions
of the Banach tensor products.
%           $$ * * *$$
We consider, mainly, the tensor norms on the tensor products of the kind
$ X^*\otimes Y.$
In this case, the tensor product $ X^*\otimes Y$
can be identified naturally with the linear space of all
finite dimensional
operators from $ X$ to $ Y$.
%In the general case, the tensor product $ X\otimes Y$
%mozhno predstavljat' sebe kak linejnoe prostranstvo vsekh
%?
%$ weak
%^*$-$to$-$weak$ nepreryvnykh konechnomernykh
%linejnykh otobrazhenij iz $ X^*$
%v $ Y$.
On the class $ {\frak T}$ of all such tensor products there is a
maximal (the strongest)
and a minimal (the weakest) tensor norms
[3]. The strongest tensor norm $ \nu_1^0$ on $ X^*\otimes Y$
generates (after completion with respect to this norm) the
{\it projective tensor product of Grothendieck} $ X^*\widehat\otimes Y$ [3],
and the weakest one --- $ \|\cdot\|$ --- {\it injective tensor product}
$ X^*\widehat{\widehat\otimes} Y$, which can be considered as the completion
of the linear space of all finite dimensional operators from $ X$ to $ Y$,
equipped with the usual operator norm $ \|\cdot\|.$
Thus, $ X^*\widehat{\widehat\otimes} Y$ can be identified with
the closed linear  subspace of
$ \operatorname{L}(X,Y)$.
Therefore, for any tensor norm $ \alpha$  between
$\nu^0_1$ and $||\cdot||,$
the natural mapping
$ X^*\otimes_{\alpha} Y\to \operatorname{L}(X,Y)$
can be extended to the
 {\it canonical map} from
$ X^*\widehat\otimes_{\alpha} Y$ to $\operatorname{L}(X,Y)$.
%Analogichno v obshchem sluchae proizvedenij vida $ X\otimes Y$
%estestvennoe otobrazhenie
%$ X\otimes_{\alpha} Y\to \operatorname{L}(X,Y)$
%prodolzhaetsja po nepreryvnosti
%do kanonicheskogo otobrazhenija iz
%$ X\widehat\otimes_{\alpha} Y$ v $\operatorname{L}(X^*,Y)$,
%prichem obraz etogo otobrazhenija lezhit v podprostranstve vsekh
%$ weak^*$-$to$-$weak$-nepreryvnykh operatorov iz $ X^*$
%v $ Y$.
%%%%%%========================

%Introduce one more notation: if $ [\operatorname{ A},a]$ is
%a (quasi)normed
%operator ideal, then
%$Z^*\widehat{\widehat\otimes}_{a} W$ is the closure of the set of all
%finite dimensional operators
%in the space  $ \operatorname{ A}(Z,W)$.

Let us give the main examples of the tensor products we will working with.
%(nekotorye tenzornye normy
%budut opredeleny nizhe).

{\it The finite $p$-nuclear tensor norm}
$\|\cdot\|_p$  for
$p\in [1,+\infty]$ is defined on the product $X\otimes Y$ by
the following way:
if $z\in X\otimes Y$, then
$$
\|z\|_p:= \inf
\biggl(\;\sum\limits_{k=1}^N
\|x_k\|^p\biggr)^{1/p}\!
\sup_{\|y'\|\leqslant 1}
\biggl\{ \left(\sum\limits_{k=1}^N
|\langle y_k,y'
\rangle |^{p'}\right)^{1/p'}\biggr\},
\tag{**}
$$
where $\ 1/p+1/p'=1\ $ and
the infimum is taken over all representations of the tensor element
 $z$ in the space  $X\otimes Y$ in the form
$
z= \sum\limits_{k=1}^Nx_k\otimes y_k$ (formally,
(**) has sense only for finite exponents
$p>1$, and for the case  $p=1$ and $p=+\infty,$
the definition have to be modified).
The completion of the tensor product $ X\otimes Y$
with respect to the norm $ \|\cdot\|_p$,
$ 1\le p\leqslant \infty$, is denoted by $ X\widehat\otimes_p Y$.
When $ p\in[1,\infty]$, the completion of the tensor product
$ X\otimes Y$ with respect to the dual to $ \|\cdot\|_p$ norm
is denoted by $ X\widehat\otimes^p Y$.

If $ p\in[1,\infty]$,
then the conjugate space to the tensor product $ X\widehat\otimes_p Y$
is equal to $ \Pi_{p'}(Y,X^{**})$
(with the natural duality defined by trace),
and
the conjugate space to the tensor product $ X\widehat\otimes^p Y$
is identified with $ \Pi_{p'}^{d}(Y,X^{**})$
(again, the duality are defined naturally, with the help of the  trace
of the superposition of a tensor element and the operator which generates
a functional)
%%%%%%%%===================
%For every  $ p\in [1,\infty],$ the norm that is dual to a tensor norm
%$ \|\cdot\|_p,$ generates a new Banach tensor product
%which is denoted by $ \widehat\otimes^p$.
All the tensor products under consideration are closely related
to corresponding operator ideals.
For example, for any Banach space
 $ X$
and
$Y$ and for each $ p\ge1,$
 the space
$ \operatorname{N}_p(X,Y)$ is a factor-space of the tensor product
$ X^*\widehat\otimes_p Y$
and
$ \operatorname{N}^p(X,Y)$ is a factor-space of the tensor product
$ X^*\widehat\otimes^p Y$.

%%========
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TXT file ended here!!!!!!

	       \bigp

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\centerline{\bf \S 2. Topological aspect of $\pi$ approximation.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	 \small


Now, let us consider in the space $ \Pi_p^d(Y,X)$
a {\it topology $ \tau_p^d$\ of
$ \pi_p^d$\!-compact convergence}, a local base of which (in zero)
is defined by sets of type
$$ \omega_{K,\e}= \left\{ U\in \Pi_p^d(Y,X):\ \pi_p^d(U\Phi_K)<\e\right\},
$$
where $ \e>0,$\, $ K=\ove{\Gamma(K)}$ is a compact absolutely convex
subset of $ Y.$



\proclaim {\bf Theorem 1}\it
Let $ \R$ be a linear subspace in
$ \Pi_p^d(Y,X),$  containing
$ Y^*\ot X.$ Then
$ (\R,\tau_p^d)'$ is isomorphic to a factor space of the space
$ X^*\wh\ot^{p'} Y.$
More precisely, if $ \ffi\in (\R, \tau_p^d)',$
then there exists
an element $ z=\sum_1^\infty x'_n\ot y_n\in X^*\wh\ot^{p'}Y$
such that
$$ \ffi(U)= \tr\, U\circ z,\ \, U\in \R. \tag$*$
$$
On the other hand, for every
$ z\in X^*\wh\ot^{p'} Y$ the relation
$ (*)$ defines a linear continuous functional on
$ (\R,\tau_p^d).$
\endproclaim\rm

{\it Proof.}\ Let
$ \ffi$ be a linear continuous functional on $ (\R,\tau_p^d).$
Then one can find a neighborhood of zero
$ \omega_{K}=\omega_{K,\e},$ such that $ \ffi$ is bounded on it:
$ \forall\, U\in\omega_K, \ |\ffi(U)|\lee 1.$
We may assume that $ \e=1.$ Consider the operator
$ U\Phi_K:\,Y_K\ovs{\Phi_K}\longrightarrow  Y
               \ovs{U}\longrightarrow X.$
Since the mapping  $ \Phi_K$ is compact,  $ U\Phi_K\in \QN_p^d(Y_K,X).$
Put $ \ffi_K(U\Phi_K)=\ffi(U)$ for $ U\in \R.$
On the linear subspace
$ \R_K= \left\{ V\in \QN_p^d(Y_K,X):\ V=U\Phi_K \right\}$
of the space  $ \QN_p^d(Y_K,X),$
the linear functional
$ \ffi_K$
is bounded: if
$ V=U\Phi_K\in \R_K$ and $ \pi_p^d(V)\lee 1,$
then $ |\ffi_K(V)|=|\ffi(U)|\lee 1.$
Therefore, $ \ffi_K$ can be extended to a linear continuous functional
$ \wt\ffi$ on the whole  $ \QN_p^d(Y_K,X);$
moreover, because of the surjectivity of the ideal
$ \QN_p^d,$
considering $ Y_K$ as a factor-space of some space $ L=l_1(\Gamma),$
we may assume that $ \wt\ffi\in\QN_p^d(L,X)^*.$ Let us mention that
$$ \wt\ffi(U\Phi_Kl)=\ffi_K(U\Phi_K)=\ffi(U) \tag1
$$
(here $ l$ is a factor map from $ L$ onto $ Y_K).$


Furthermore, since
$ \QN_p^d(L, X)^*= \I^{p'}(X, L^{**}),$
we can find an operator
$ \Psi: X\to L^{**},$\,
$\Psi: X \ovs{A}\longrightarrow L_p \ovs{i}\longrightarrow L_1
\ovs{B}\longrightarrow L^{**} $
(here $i$ is an injection),
for which
$$ \wt\ffi(A)= \tr \Psi A,\ \, A\in L^*\ot X.
$$

Let
$ A_n\in L^*\ot X,\, $ $ \pi_p^d(A_n-U\Phi_Kl)\to 0.$
Then
$$ \wt\ffi(U\Phi_Kl)= \lim\, \tr \Psi A_n. \tag2
$$

Consider the operator
$ \Phi_K^{**}l^{**}\Psi :
  X\ovs{A}\longrightarrow  L_p \ovs{i}\longrightarrow L_1
  \ovs{B}\longrightarrow L^{**} \ovs{l^{**}}\longrightarrow (Y_K)^{**}
\ovs{\Phi_K^{**}}\longrightarrow Y.$
Since
$ i\in \I^{p'}$ and $ \Phi_K$
is compact then
$ \Phi_K^{**}l^{**}\Psi  \in X^*\wh\ot^{p'} Y.$
% Ho!: go via 2 compact operators and refl. space by Johnson!
Let
$ \sum_{n=1}^\infty \mu_n\ot y_n$
be a representation of the operator
% $iBl^{**}\Phi_K^{**)$
$\Phi_K^{**}l^{**}Bi $ in
$ L_p^*\wh\ot^{p'}Y.$
Put
$ z=\sum A^*(\mu_n)\ot y_n.$
The element
$ z$
generates an operator
$ \Phi_K^{**}l^{**} \Psi $
from
$ X$
to
$ Y.$
We will show that
$ \tr U\circ z=\wt\ffi(U\Phi_K l)$
(note that
$ U\circ z$
is an element of the space
$ X^*\wh\ot X,$
so the trace is well defined).
We have:
$$\multline
 \tr U\circ z= \tr \( \sum A^*(\mu_n)\ot Uy_n\)=
   \sum \< A^*(\mu_n), Uy_n\> =\\ =
\sum \< \mu_n, AUy_n\>=
\tr AU\Phi_K^{**}l^{**}Bi =\tr (U\Phi_K)^{**}l^{**}\Psi,
\endmultline                                     \tag3
$$
where
$ (U\Phi_K)^{**}l^{**}\Psi:
    X\ovs{\Psi}\longrightarrow L^{**}
       \ovs{l^{**}}\longrightarrow Y^{**}
         \ovs{\Phi_K^{**}}\longrightarrow Y
       \ovs{U}\longrightarrow X.$
% identify X with its image in  X**; come through with a compact act!!!!
Since
$ \pi_p^d(A_n- U\Phi_Kl)\to 0,$
then
$ \pi_p^d\(A^{**}_n - (U\Phi_Kl)^{**}\)\to 0.$
Moreover, if
$A_n=\sum_1^N w_m\ot f_m\in L^*\ot X,$
then
$$ \tr A_n^{**}\Psi= \sum_m \< \Psi^*w_m, f_m\> =
   \sum_m \< w_m, \Psi f_m\>= \tr \Psi A_n.
$$

Hence,
$ \tr (U\Phi_Kl)^{**}\Psi = \lim \tr A^{**}_n\Psi=
\lim \tr \Psi A_n.$
Now, it follows from (3) and (2) that
$ \wt\ffi(U\Phi_Kl)=\tr U\circ z.$
Finally, from (1):
$ \ffi(U)= \tr U\circ z.$
Thus, the functional
$ \ffi$
is defined by an element of
$ X^*\wh\ot^{p'} Y.$

Inversely, if
$ z\in X^*\wh\ot^{p'} Y,$
put
$ \ffi(U)=\tr U\circ z$
for
$ U\in \R$
(the trace is defined since
$ U\circ z\in X^*\wh\ot X).$
We have to show that the linear functional
$ \ffi$
is bounded on a neighborhood
$ \omega_{K,\e}$
of zero in
$ \tau_p^d.$
For this, we need

\proclaim {\bf Lemma 1}\it
If
$ z\in X^*\wh\ot^q Y,$
then $ z\in X^*\wh\ot^q Y_K,$
where
$ K=\ove{\Gamma(K)}$
is a compact in
$ Y.$
\endproclaim\rm

\demo{\it Proof of the lemma
}\
Let
$ z=\sum x'_n\ot y_n,$ $ \{c_n\}\in c_0$
and
$ \sum \|y_n\|^q\,c_n^{-1}<+\infty. $
Consider the operator
$ A_1:l_{q'}\to l_1,$\,
 $ A_1\{a_n\}= \sum a_n c_n^{-1} \,||y_n||\, e_n$ \,
	     ($ e_n$ are orths in $ l_1).$
Let $A_0: l_1\to Y,$\ $A_0\{b_n\}=\sum b_n c_n ||y_n||^{-1}y_n.$
Since this operator is compact, one can find a compact
$ K\sbs Y$
and an operator
$ A_2: l_1\to Y_K,$
for which
$ A_0=\Phi_K A_2.$
Put
$ z_0=\sum x'_n\ot e_n$\, ($ e_n$ are orths in $ l_q).$
Then
$ \ove{z}:=(\bold 1\ot [A_2A_1])(z_0)\in X^*\wh\ot^q Y_K,$
and
$ \Phi_K(\ove{z})=z.$
 \enddemo

Let us continue the proof of Theorem. Let
$ K$ be a compact in
$ Y,$  with
$ z\in X^*\wh\ot^{p'} Y_K.$
If
$ U\in \omega_{K,1},$
then
$ \pi_p^d(U\Phi_K)<1$
and
$ |\tr U\Phi_K\circ z|\lee
\|z\|_{X^*\wh\ot^{p'} Y_K}\,\pi_p^d(U\Phi_K)\lee C.$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\proclaim {\bf Corollary 1}\it
$ (\R,\tau_p^d)'=(\R,\sigma)',$
where
$ \sigma=\sigma(\R, X^*\wh\ot^{p'} Y).$
Thus, the closures
of convex subsets of the space
$ \Pi_p^d(Y, X)$
in
$ \tau_p^d$
and in
$ \sigma$
are the same.
 \endproclaim\rm

Denote by
$ X^*\wt\ot^{p'} Y$
the closure of the set
$ X^*\ot Y$
in the space
$ \I^{p'}(X, Y^{**})$
(dual to
$ Y^*\wh{\wh\ot}^p X$ --- the closure of $Y^*\ot X$
in the space $\QN_p^d(Y,X)$)

\proclaim {\bf Theorem 2}\it
Let
$ A$
be the intersection of the unit ball of the
space
$ G=G(Y, X^{**})$\,
(dual to
$ X^*\wt\ot^{p'} Y)$\,
with the subspace
$ Y^*\ot X.$ $ {*}$\!-weak
closure of the set
$ A$ in $ G\cap \Pi_p^d(Y,X)$
coincides with the closure of
$ A$
in
$ (\Pi_p^d(Y, X), \tau_p^d).$
\endproclaim\rm

\demo{\it
Proof
}\
Let us consider the canonical mappings
$$ \CD
  X^*\wh\ot^{p'} Y  @>j>>  X^*\wt\ot^{p'} Y, \\
\Pi_p^d(Y, X^{**})    @<j^*<<  G(Y, X^{**}).
\endCD
$$

Since
$ j^*$
is one-to-one, then the closures of the bounded sets in
$ G(Y, X^{**}),$
in topologies
$ \sigma(G, X^*\wt\ot^{p'} Y)$
and
$ \sigma (G, X^*\wh\ot^{p'} Y),$
are the same. Therefore, if
$ B$ is a convex bounded set in
$ G\cap \Pi_p^d(Y,X),$
then the closure of the set
$ B$
in
$ \( \Pi_p^d(Y,X)\cap G, \sigma(G, X^*\wt\ot^{p'} Y)\)$
coincides with the closure of the set
$ B$
in the space
$ \( \Pi_p^d(Y,X), \sigma(\Pi_p^d(Y,X), X^*\wh\ot^{p'} Y)\)$
and therefore,
by Corollary 1, with the closure of
$ B$
in
$ \( \Pi_p^d(Y,X), \tau_p^d\).$
\enddemo

\proclaim {\bf Corollary 2}\it
With notations of the theorem {\rm 2},
the closure of the set
$ A$
in
$ \tau_p$
coincides with the closure of
$ A$
in the space
$ \L(Y,X)$
in the topology of compact convergence.
\endproclaim\rm
For the {\it proof}\ it is enough to use the previous assertion, considering
the canonical mapping from
$ X^*\wh\ot_1 Y$
into
$ X^*\wt\ot^{p'} Y$
instead of the map
$ j$
from the proof of the theorem 2
(and to apply either Theorem 1
for
$ p=+\infty,$
or results on dualities from [3]).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!!!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!!!
\proclaim {\bf Corollary 3}\it
Let
$ C>0$
and
$ T\in \Pi_p^d(Y,X).$
The following assertions are equivalent:

$1)$
there ia a net
$ \left\{ T_\al\right\}, T_\al\in Y^*\ot X,$
converging to
$ T$
in the topology
$ \tau_p^d$
such that
$ \pi_p^d(T_\al)\lee C;$

$2)$
there is a net
$ \left\{ T_\al\right\}, T_\al\in Y^*\ot X,$
converging to
$ T$
in the topology of compact convergence,
such that
$ \pi_p^d(T_\al)\lee C;$

$3)$
the canonical mapping
$$ \CD
  X^*\wh\ot^{p'} Y  @>j>>  X^*\wt\ot^{p'} Y
\endCD
$$
is $C$-isometric.
\endproclaim\rm




\proclaim {\bf Theorem 3}\it
For an operator
$ T\in \Pi_p^d(Y,X), \ove{T(Y)}=X,$
the following are the same:

$1)$ $ T\in \ove{Y^*\ot X}^{\,\tau_p};$

$2)$
there is a net of operators
$ R_\al\in Y^*\ot Y$
such that
$ TR_\al\to T$
in the topology
$ \tau_p^d.$
\endproclaim\rm

\demo{\it Proof}
Assuming that 2) is not valid, we (by Corollary 1) get:
$$ \not\exists \, R_\al\in Y^*\ot Y:\ TR_\al\to T\ \text{ in }
   \( \Pi_p^d(Y,X), \sigma(\Pi_p^d(Y,X), X^*\wh\ot^{p'} Y)\).  \tag4
$$

Consider the associated with
$ T$
mappings:
$$ \CD
  X^*\wh\ot^{p'} Y @>\wt{T}>>   Y^*\wh\ot_1 Y, \\
   \Pi_p^d(Y, X^{**})  @<\wt{T}^{\,*}<<  \L(Y, Y^{**}).
\endCD
$$
where
$ \wt{T}(z)= z\circ T$
for
$ z\in X^*\wh\ot^{p'} Y.$
Let
$ Z=\ove{\wt{T}^{\,*}(Y^*\ot Y)}^{\,*}$
(the closure is taken in the space
$ \Pi_p^d(Y,X)$
in
$ {}^*$\!-weak topology).
It follows from (4) that
$ T$
is not zero on the subspace
$ Z^{\perp}\sbs X^*\wh\ot^{p'} Y,$
i.e. there exists an
$ A\in Z^{\perp}$
such that
$ \< T,A\>=\tr AT=1.$  But $ AT=\wt{T}(A),$
and if
$ R\in Y^*\ot Y,$ then $ \< \wt{T}(A), R\>=
\< A, (\wt T)^{\,*}(R)\>=0.$
Hence, the element
$ \wt{T}(A)$
of the projective tensor product
$ Y^*\wh\ot Y$
is not zero
(since
$ \tr \wt{T}(A)=1),$
but generates a null-operator in $ Y.$
For any
$ y'\in Y^*$
and
$ Ty\in T(Y)$
we have:
$ \< A, Ty\ot y'\>= \< ATy,y'\>=0.$
Since
$ \ove{T(Y)}=X,$
we obtain that a non-zero tensor element
$ A\in X^*\wh\ot^{p'} Y$
which generates a zero-operator. Again, by using the equality
$ \tr AT=1,$
we conclude that
$ T$
can not be approximated in
$ {^*}$\!-weak
topology by finite rank operators.
Now, it follows from Corollary 1 that the condition 1)
is not fulfilled.
 \enddemo

Next two statements give us sufficient (but not necessary, as we will see below)
conditions for the density of the set of all finite rank operators in the space
of operators
$ \Pi_p^d(Y,X)$
in the topology
$ \tau_p^d$\ of
$ \pi_p^d$-compact convergence.

\proclaim {\bf Theorem 4}\it
%If $ \QN_p(Y,X)= \ove{Y^*\ot X}^{\,\pi_p},$ then  !!!
%$ \Pi_p(Y,X)= \ove{Y^*\ot X}^{\,\tau_p}.$
% --- So was formulation in that paper what is problematic -
%  so, in the case $p=\infty$ it is obtained somewhat open question
%    Here is a more normal formulation:           !!!
If
$ \QN_p^d(Y,X)= \ove{Y^*\ot X}^{\,\pi_p^d}$
for every Banach space
$ Y,$  then for each
$ Y$\
$ \Pi_p^d(Y,X)= \ove{Y^*\ot X}^{\,\tau_p^d}.$
\endproclaim\rm

\demo{\it Proof}
Let
$ U\in \Pi_p^d(Y,X),$ $ \e>0,$ $ K=\ove{\Gamma(K)}$
be a compact in
$ Y.$
By assumptions, there is an operator
$ V\in (Y_K)^*\ot X,$
such that
$ \pi_p^d(V-U\Phi_K)<\e.$
%  To clear this - or more general formulation on product of
%  summing and RN-operators!!!
We need to change successfully $ V$
by an operator
$ \wt V\Phi_K,$
where
$ \wt V\in Y^*\ot X.$
Let
$ V=\sum_{n=1}^N z_n\ot x_n.$
Note that we can consider only the case where
$ \Phi_K^{**}$
is one-to-one
(else, with the help of the construction of [2]
we change
$ Y_K$
by a space
$ Y_{K_0},$
for which the operator
$ \Phi_{K_0}$
is compact and the operator
$ \Phi_{K_0}^{**}$
is one-to-one).
In this case
$ Y^*$  is norm dense in
$ (Y_K)^*$
and, therefore, for every positive number sequence
$ \{ \e_n\}$
there exist the elements
$ y'_n\in Y^*,$
for which
$ \|y'_n-z_n\|_{Y^*_K}<\e_n.$
Put
$ \wt V=\sum_1^N y'_n\ot x_n\in Y^*\ot X.$
Let
$ \{ a_n\}$
be a sequence of the elements of the space
$ X^*,$
such that
$ \sup \left\{ \sum | \< a_n,a'\>|^p:\ \|a'||_{X}\lee 1\right\}\lee 1.$
We have:
$$ \multline
 \sum_{i=1}^m \|(\wt V\Phi_K-V)^*a_i\|^p =
  \sum_{i=1}^m \|\sum_{n=1}^N (y'_n-z_n)\, \<a_i,x_n\>\|^p\lee \\ %\lee
  \sum_{i=1}^m \(\sum_{n=1}^N  \| y'_n-z_n\|^{p'}\)^{p/p'}
     \sum_{n=1}^N |\<x_n,a_i\>|^{p} \lee  \\ %\lee
     \sum_{n=1}^N \|x_n\|^{p}
   \(\sum_{n=1}^N \|y'_n-z_n\|^{p'}_{Y^*_K}\)^{p/p'}\lee
      \sum_{n=1}^N \|x_n\|^{p}
      \(\sum_{n=1}^N \e_n^{p'}\)^{p/p'}.
\endmultline
$$

If we take
$ \e_n$
small enough then the last number is less then
$ \e,$
and, from the inequality
$ \pi_p\((V-U\Phi_K)^*\)\lee\e,$
we get that
$ \pi_p^d(\wt V\Phi_K-U\Phi_K)\lee \e+ \pi_p^d(V-\wt V\Phi_K)\lee 2\e.$
Hence,
$ \wt V-U\in \omega_{K,2\e}.$
Thus, we have shown that for every neighborhood
$ \omega_{K,\e}$\
there exists an operator
$ \wt V\in Y^*\ot X,$
for which
$ \wt V-U\in \omega_{K,\e}.$
Therefore
$ U\in \ove{Y^*\ot X}^{\,\tau_p^d}.$
 \enddemo

\proclaim {\bf Theorem 5}\it
If the canonical mapping
$ j: X^*\wh\ot^{p'} Y\to \N^{p'}(X,Y)$
is one-to-one then
$ \Pi_p^d(Y,X)= \ove{Y^*\ot X}^{\,\tau_p^d}.$
\endproclaim\rm
\demo{\it Proof}
If the map
$ j$
is one-to-one then the annihilator
$ j^{-1}(0)^\perp$
of its kernel in the space, dual to
$ X^*\wh\ot^{p'}Y,$
coincides with
$ \Pi_p^d(Y, X^{**}).$
On the other hand, in any case
$ j^{-1}(0)^\perp=
\ove{Y^*\ot X}^{\,*}$ (
the closure in
$ {}^*$\!-weak topology of the space
$ \Pi_p^d(Y, X^{**}));$
by Corollary 1,
$$ \Pi_p^d(Y,X)\cap \ove{Y^*\ot X}^{\,*}= \ove{Y^*\ot X}^{\,\tau_p^d}.
$$

Therefore,
$ \Pi_p^d(Y,X)= \ove{Y^*\ot X}^{\,\tau_p^d}.$
 \enddemo

For a reflexive space
$ X,$
the dual space to
$ X^*\wh\ot^{p'} Y$
is equal to
$ \Pi_p^d(Y,X).$
Consequently, by Corollary 1 and Theorem 5,
we get

\proclaim {\bf Corollary 4}\it
For a reflexive pace
$ X$
the canonical mapping
$ j: X^*\wh\ot^{p'} Y\to \N^{p'}(X,Y)$
is one-to-one iff the set of finite rank operators is dense in the space
$ \Pi_p^p(Y,X)$
in the topology
$ \tau_p^d$\  of $ \pi_p^d$-compact convergence.
\endproclaim\rm

To conclude this part of our considerations,
let us give an assertion which shows that
it  follows from the existing
of an dually absolutely $p$-summing operator, non-approximated in the topology
$ \tau_p^d,$
the existing of  a non-approximated in the same topology dually
quasi-$p$-nuclear operator
(with values in the same space).

\proclaim {\bf Theorem 6}\it
If there exists an operator
$ T\in \Pi_p^d(Y,X)\setminus \ove{Y^*\ot X}^{\,\tau_p^d},$
then there exist a reflexive space
$ Z$
and an operator
$ U\in\QN_p^d(Z,X),$
which is not in the closure
$\ove{Y^*\ot X}^{\,\tau_p^d}.$
\endproclaim\rm
For the  {\it proof},\
it is enough to remember the definition of the topology $ \tau_p^d$
and to use the following two facts:

a)
if
$ V$
is a compact operator then
it can be represented as a composition of two compact operators;

b)
the product
$ AB$
of a compact operator
$B$
and dually absolutely $ p$-summing operator
$ A$
is a dually quasi-$p$-nuclear map
(concerning a), see [4] ;
b) is a evident).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !!!


	     %----\S2 in App_85

{
Now, we fix the space of images of operators (or the spaces where operators are defined),
and investigate the conditions under which it is possible
to approximate (by finite rank operators) {\it all}\ dually
absolutely $p$-summing mappings
with values in a given space (or acting from a given space).
				     }

The next statement, among other things, gives us a partial inversion of
Theorem 4.

\proclaim {\bf Theorem 7}\it
For a Banach space
$ X$
the following are equivalent:

$1)$
for every Banach space
$ Y$
the equality
$ \QN_p^d(Y,X)= \ove{Y^*\ot X}^{\,\pi_p^d}$
holds;

$2)$
for every Banach space
$ Y$
the equality
$ \Pi_p^d(Y,X)= \ove{Y^*\ot X}^{\,\tau_p^d}$
holds;

$3)$
for every Banach space
$ Y$
the equality
$ \QN_p^d(Y,X)= \ove{Y^*\ot X}^{\,\tau_p^d}$
holds;

$4)$
for every reflexive Banach space
$ Y$
the equality
$ \QN_p^d(Y,X)= \ove{Y^*\ot X}^{\,\tau_p^d}$
holds.
\endproclaim\rm

\demo{\it Proof}
Implications
$ 2)\implies 3)\implies 4)$
are evident;
$ 1)\implies 2)$
by Theorem 4.
For the proof of the implication
$ 4)\implies 1),$
consider an operator
$ V\in \QN_p^d(Y,X).$
By using the results of
[2],
 we can factorize the operator
$ V$
by the following way:
$ V=UA,$
where
$ A\in \operatorname{K}(Y,Z)$
(compact),
$ U\in \QN_p^d(Z,X),$
and, moreover, the space
$ Z$ is reflexive. Put
$ K=\ove{A({B}_Y)}.$
From 4), it follows that
$$ \forall\,\e>0\ \exists\, \wt V\in Z^*\ot X:\ \ \wt V-U\in\omega_{K,\e},
$$
i.e.
$ \pi_p^d(\wt V\Phi_K-U\Phi_K)<\e.$
Hence,
$ \pi_p^d(V-\wt VA)\lee C_A\,\pi_p^d(U\Phi_K- \wt V\Phi_K)\lee C_A\e.$
 \enddemo

\proclaim {\bf Corollary 5}\it
If for every reflexive space
$ Y$
the canonical mapping
$ X^*\wh\ot^{p'} Y \to \N^{p'}(X,Y)$
is one-to-one then for each Banach space
$ Y$
the equality
$ \QN_p^d(Y,X)= \ove{Y^*\ot X}^{\,\pi_p^d}$
holds.
 \endproclaim\rm

For the  {\it proof},\  Theorem 5 can be applied; then use
the implication
$ 2)\implies 1)$
of the previous result.


It follows from Corollaries 4 and 5 and Theorem 7.

\proclaim {\bf Corollary 6}\it
For a reflexive Banach space
$ X$
the following are equivalent:

$1)$
for each space
$ Y$
the canonical mapping
$ X^*\wh\ot^{p'} Y \to \N^{p'}(X,Y)$
is one-to-one;

$2)$
for each space
$ Y$
the set of finite rank operators is dense in the Banach space
$ \QN_p^d(Y,X).$
 \endproclaim\rm

The next statement, among other things, gives us a partial inversion of
Theorem 5.

\proclaim {\bf Corollary 7}\it
For every Banach space
$ X$
the following are equivalent:

$1)$
for each space
$ Y$
the canonical mapping
$Y^*\wh\ot^p X \to \N^p(Y,X)$
is one-to-one;

$2)$
for each reflexive Banach space
$ Y$
the canonical mapping
$Y^*\wh\ot^p X \to \N^p(Y,X)$
is one-to-one;

$3)$
for each (reflexive) Banach space
$ Y$\
$ \ove{X^*\ot^{p'} Y}^{\,\tau_{p'}^d}= \Pi_{p'}^d(X,Y).$
\endproclaim\rm
\demo{\it Proof}\,
Concerning the equivalence
$ 1)\iff 2),$
see [18];
the implications
$ 1)\implies 3)$
and
$ 3)\implies 2)$
follow from Theorem 5 and Corollary 4, respectively.
 \enddemo


The previous statement yields the well known result:

\proclaim {\bf Corollary 8 {\rm (P.\, Saphar, O.I. Reinov)}}\it
If a Banach space
$ X$
has the AP then for any
$ p>1$
and any space
$ Y$
the canonical mapping
$Y^*\wh\ot^p X \to \N^p(Y,X)$
is one-to-one.
 \endproclaim\rm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !!! !!!
The above results lead us to the following definition which is equivalent to
corresponding definition, e.g., in [17],[18].


\definition {\bf Definition 1}
Let
$ p\gee 1.$
A Banach space
$ X$
has the property
$ \AP^p$
(the approximation property of order
$ p^{dual}$),     %%%%%%%%%%%%  ???
if every dually absolutely
$ p'$-summing operator, acting from the space
$ X,$
can be approximated in the topology of
$ \pi_{p'}^d$-compact convergence
$ \tau_{p'}^d$
by operators of finite rank.
\enddefinition

It follows from
Theorem 4 and Corollary 7
 that

\proclaim {\bf Corollary 9}\it
If for each Banach space
$ Y$
the equality
$ \QN_{p'}^d(X,Y)=\ove{X^*\ot Y}^{\,\pi_{p'}^d}$
holds then the space
$ X$
has the property
$ \AP^p$
in the sense of Definition $1.$
 \endproclaim\rm

 \comment
Recall for the sake of completeness the following
assertion on a characterization of the spaces with the property
$ \AP^p$
(a proof can be found in
[57]).    %%%%%%%% ???

\proclaim {\bf Theorem 8}\it    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ???
A Banach space
$ X$
has the property
$ \AP^p$
in the sense of Definition $1$
iff
for every Banach space
$ Y,$
for every operator
$ T\in \Pi_{p'}^d(X,Y),$
for each weakly
$ p'$-summable sequence
$ \{ x_k\}$
of elements of the space
$ X$
and for every
$ \e>0$
there is a finite rank operator
$ R: X\to Y,$
such that
$ \sum \|Ux_k-Rx_k\|^{p'}<\e.$
\endproclaim\rm

   \endcomment
	%!!! From survey p31 oldest

        {
Now, let us consider some counterexamples
in connection with the theme of our investigations
(a lot of counterexamples concerning AP's
can be found in [5], [7], [11--20].
Bellow, we make use of  results of [12], [18].
Namely:
for any
$ q\gee 1,$ $ q\neq2,$
there exists a separable reflexive Banach space without the property
$ \AP^q.$
Moreover, for every
$ q\gee 1,$ $ q\neq2,$
there exist a separable reflexive
$ E,$
an operator
$ R\in \Pi_{q'}^d(E,E)$
and a tensor element
$ t\in E^*\wh\ot^q E,$
so that
$ \tr t\circ R=1$ and $ \tr t\circ A=0$
for each finite rank operator
$ A\in E^*\ot E.$
					 }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  !!! 67-8


It is often very useful to apply the following fact when
constructing some counterexamples
([6]):
\it
For every separable Banach space
$ E$
there exist a separable conjugate Banach space
$ H=Y^*$
with a basis and operators
$ Q: H\to E$
and
$ U: H^*\to E^*$
so that
$ Q(H)=E,$
$ U(H^*)=E^*,$ $ \|U\|\lee 1,$ $ \|Q\|\lee 1,$ $ UQ^*=\id_{E^*}$
and the space
$ \( \id_{E^*}- Q^*U\)(H^*)$
is isomorphic to the space
$ Y.$
\rm

Now we are ready for constructions of our counterexamples.
Firstly, we will show that the inversion of
Theorem 5 and Corollary 5
are not valid.

\proclaim {\bf Theorem 8}\it
For every
$ p\in [1,+\infty],$ $ p\neq2,$
there exist a separable reflexive space
$ E,$
a separable conjugate space
$ H$
with a basis such that the canonical mapping
$ j: H^*\wh\ot^{p'} E\to \N^{p'}(H,E)$
is not one-to-one. On the other hand, since
$ H$
has the Grothendieck AP,
$ \QN_p^d(E,H)=\ove{E^*\ot H}^{\,\pi_p^d}$
and
$ \Pi_p^d(E,H)= \ove{E^*\ot H}^{\,\tau_p^d}.$
\endproclaim\rm

\demo{\it Proof}
Let
$ E,$ $ t\in E^*\wh\ot^{p'} E$
and $ R\in \Pi_p^d(E,E)$
be the mentioned above spaces, tensor element and operator so that
$ \tr t\circ R=1$
and
$ t=0$
as an operator.
Set
$ g=t\circ Q\in H^*\wh\ot^{p'} E$
and
$ V=U^*R\in \QN_p^d(E, H^{**}).$
Then
$ \tr V\circ g=1$
(consequently, the tensor element
$ g$
is not equal to zero)
and
$ g=0$
as an operator.
 \enddemo

\proclaim {\bf Corollary 10}\it
For every
$ p\in[1,+\infty],$ $ p\neq2,$
there exist a Banach space
$ Z,$
an element
$ z\in Z^*\wh\ot^{p'} Z$
and an operator
$ \Psi\in\Pi_p^d(Z,Z^{**})$
such that
$ \tr \Psi\circ z=1,$
but
$ \tr \Phi\circ z=0$
for all
$ \Phi\in \Pi_p(Z,Z).$
\endproclaim\rm
\demo{\it Proof}\,
Let us use notation introduced in the proof of
Theorem 8.
Let
$ \sum y'_n\ot y_n$
be any representation of a tensor element
$ g\in H^*\wh\ot^{p'} E.$
Note that
$ \tr A\circ g=0$
for every operator
$ A\in \Pi_p^d(E,H)$
(because of the space
$ H$
has AP).
Put
$ Z=H\oplus E,$
$ z=\sum (y'_n, 0)\ot (0,y_n)$
and define the operator
$ \Psi\in\QN_p^d(Z,Z^{**})$
by
$ \Psi(h,y)=(Vy,0).$
We have:
$$ \tr \Psi\circ z=\sum \< (y'_n,0), (Vy_n, 0)\>=
     \sum \< y'_n, Vy_n\> =\tr V\circ g=1.
$$

On the other hand, denoting by
$ P_H$
and
$ P_E$
the natural projectors from
$ Z$
onto
$ H$
and
$ E$
respectively, we have, for arbitrary operator
$ \Phi\in \Pi_p^d(Z,Z),$

$$ \Phi(h,y)=\Phi|_H(h)+\Phi|_E(y)=
   \( P_H\Phi|_H(h)+P|_E\Phi|_H(h)\)+
   \( P_H\Phi|_E(y)+P|_E\Phi|_E(y)\),
$$
whence,
$$ \tr \Phi\circ z= \sum \< (y'_n, 0), \[ P_H\Phi(0, y_n)\]\>=
	\sum \< (y'_n,0), P_H\Phi|_E(y_n)\>.
$$
Denoting by
$ A$
the operator
$ P_H\Phi|_E,$
we get:
$$ A\in \Pi_p^d(E,H);\ \ \tr \Phi\circ z=\tr A\circ g=0.
$$
\enddemo

\remark {\bf Remark 1}
For
$ p=+\infty,$
we get again (cf. [15]) nonzero tensor element
$ z\in Z^*\wh\ot_1 Z,$
vanishing on the subspace
$ \L(Z,Z)$
of the space
$ \L(Z,Z^{**}).$
This is an answer to a question of Swedish mathematician
Sten Kaijser.
\endremark\medpagebreak

Now we will show that the inversion of
Theorem 4 and Corollary 9 are invalid too.

\proclaim {\bf Theorem 9}\it
For every
$ p\in [1,+\infty],$ $ p\neq2,$
there exist a separable reflexive space
$ E,$
a separable conjugate space
$ H$
with a basis
$($so, with the property
$ \AP^{p'})$
such that
$ \QN_{p}^d(H,E)\neq \ove{H^*\ot E}^{\,\pi_p^d};$
on the other hand,
$ \Pi_p^d(H,E)=\ove{H^*\ot E}^{\,\tau_p^d}$ \, {\rm (see Corollary 7).}
\endproclaim\rm
\demo{\it Proof}
With the notation of
Theorem 8,
set
$ L=RQ.$
Since
$ \tr (t\circ RQ^{**}Y^*)=1$
and $ t=0$
as an operator,
the map
$ RQ^{**}$
can not be approximated by finite rank operators in the space
$ \QN_p^d(H^{**},E).$
Moreover,
$ L\not\subset \ove{H^*\ot E}^{\,\pi_p^d}.$
 \enddemo

We finish the paper with an application of the above results to a
solution of A. Grothendieck's problem, posed him in
[3, Chap. 2, p. 135, ``Question non R\'esolues"].
A. Grothendieck had showed that for Banach spaces with weakly compact
 identity maps (for reflexive spaces) the approximation
and the bounded approximation properties are the same.
 This led him to the following question:\,
 {\it whether any weakly compact operator with AP possesses also BAP}?
  As to this question on weakly compact operators, it is
a paper of mine [14], where the first counterexample
to this problem was constructed. There was shown ("by hands")
that there is a compact operator (from a Banach space with AP), that
can not be approximated, in the topology of the compact convergence,
by finite rank operators with uniformly bounded norms.

The next striking enough result (proved early by me by using
a rather different and
difficult method)
shows that such operators may be lie
even in the classes of dually quasi-$p$-nuclear operators (which are
evidently compact).

For the proof, we need the following assertion from my works [17] and
[18, Theorem 4.21,(1)]: there exist a separable space $E$ with the property
$\AP,$ a separable reflexive space $F$ such that for every $q\in[1,2)$
the canonical mapping
$$ \CD
  F^*\wh\ot^{q} E  @>j>>  F^*\wt\ot^{q} E
\endCD
$$
is one-to-one but is not $C$-isometric, for any $C>0.$

\proclaim {\bf Theorem 10}\it
There exist a separable space $Y$ with the property
$\AP,$ a separable reflexive space $X$ such that for every $p\in(2,+\infty]$
there exists an operator
$ T\in \QN_p^d(Y,X)$ for which:
  \roster
%\item "{}"
for any constant $C>0,$ there is no net
$ \left\{ T_\al\right\}, T_\al\in Y^*\ot X,$
converging to
$ T$
in the topology of compact convergence,
and such that
$ \pi_p^d(T_\al)\lee C;$
\endroster
\endproclaim\rm
\demo{\it Proof}
Put $X:=F,$ $Y:=E$ and $p=q'$
and apply just mentioned result from [18]
and Corollary 3 (note that for a reflexive $X$ one has
$\Pi_p^d(Y,X)=\QN_p^d(Y,X);$ see [8], [18]).

\enddemo




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\bigp

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%\newpage

\Refs


\ref \no1 \by Bourgain J., Reinov O.I. \pages 19-27
\paper On the approximation properties   for the space $H^\infty$
\yr 1985\vol   122
\jour Math. Nachr.
\endref


\ref \no2 \by  Davis W.J, Figiel T., Johnson W.B., Pelczynski A.
                             \pages     311-327
\paper    Factoring weakly compact operators
\yr 1974 \vol 17 \issue
\jour J. Functional Analysis
\finalinfo     $MR 50 \# 8010.$
\endref


\ref \no3 \by Grothendieck A. \pages pp. 196 + 140
\paper  Produits tensoriels topologiques et espases nucl\'eaires
\yr 1955\vol  16
\jour  Mem. Amer. Math. Soc.
\finalinfo
\endref


\ref \no4
\by Johnson W.B. \pages 337-345
\paper  Factoring compact operators
\yr 1971 \vol 9 \issue
\jour         Israel J. Math.
\endref


\ref \no5
\by Kaijser Sten, Reinov Oleg
\pages 59--64
 \paper  On $\alpha$-nuclearity and total accessibility for some tensor norms $\alpha$
 \yr 2001 \vol  5
 \jour Acta Comment. Univ. Tartu. Math.
 \endref


\ref \no6 \by Lindenstrauss J.\pages  279-284
\paper  On James' paper "Separable Conjugate Spaces"
\yr 1971\vol 9
\jour Israel J. Math.
\endref


\ref \no7 \by Oja E.,  Reinov O.I.\pages 121-122
\paper  Un contre-exemple a une affirmation de  A. Grothen\-dieck
\yr 1987\vol 305
\jour  C. R. Acad. Sc. Paris, Serie I
\endref

\ref \no8
\by Persson A. \pages 213--232
\paper  On some properties of $p$-nuclear and $p$-integral operators
\yr 1969\vol 33
\jour Studia Math.
\finalinfo
\endref

\ref \no9 \by Persson A. and Pietsch A.\pages 19-62
\paper  p-nucleare und p-integrale Abbildungen   in Banachr\"aumen
\yr 1969\vol33
\jour Studia Math.
%\finalinfo   $MR 39\#4645.$
\endref


\ref \no10
\by   Pietsch A.
\book  Operator ideals
\bookinfo North-Holland%Edited by
\publ Deutscher Verlag der Wiss. \vol \nofrills
\publaddr  Berlin
\yr  1978. 451 p.
\endref


\ref \no11 \by Reinov O.I.  \pages 43-47
\paper  Approximation properties of order p and the existence of
  non-p-nuclear operators with p-nuclear second adjoints
\yr 1981 \vol 256 \issue  1
\jour Doklady AN SSSR
\finalinfo
\endref


\ref \no12 \by Reinov O.I.\pages   125-134
\paper   Approximation properties of order p and the existence of
  non-p-nuclear operators with p-nuclear second adjoints
\yr 1982 \vol  109
\jour    Math. Nachr.
\endref


\ref \no13 \by Reinov O.I. \pages 145-165
\paper  Disappering tensor elements in the scale of p-nuclear operators
%\pages 145-165
\yr 1983\vol
\jour Theory of operators and theory of functions (LGU)
\endref

\ref \no14   \by Reinov O.I.  \pages 597-599
\paper Un contre-exemple a une conjecture de A.Grothendieck
\yr 1983 \vol 296 \issue
\jour C. R. Acad. Sc. Paris, Serie I
\finalinfo
\endref


\ref  \no15
\by  Reinov O.I.
\paper Approximation of operators in Banach space
\pages 128–142
\inbook Problems of Approximation Theory
\bookinfo
\publ  \vol 156
\publaddr Kalinin, Kalinin. Gos. Univ.
\yr  1985
\endref


\ref \no16 \by Reinov O.I.  \pages 905-907
\paper Sur les operateurs p-nucl\'eaires entre espaces de Banach avec bases
\yr 1993\vol  316
\jour  C. R. Acad. Sc. Paris. ---  Serie I
\endref


\ref  \no17
\by  Reinov O.I.
\paper Approximation properties and some classes of operators
\pages 147-205
\inbook Problems of mathematical analysis, vol. 23
\bookinfo
\publ  \vol
\publaddr Novosibirsk: Nauchn. Kniga
\yr  2001
\endref


\ref  \no18
\by  Reinov O.I.
\paper Approximation properties and some classes of operators.
Function theory and mathematical analysis
\pages 3911--3951
\inbook J. Math. Sci.
\bookinfo
\publ  \vol 112
\publaddr New York
\yr  2001
\endref


\ref  \no19
\by  Reinov O.I.
\paper How bad can a Banach space with approximation property be? II. Function theory and applications
\pages 4065--4072
\inbook J. Math. Sci.
\bookinfo
\publ  \vol 112
\publaddr New York
\yr  2002
\endref

\ref  \no20
\by  Reinov O.I.
\paper Excursus to the approximation theory of operators in operator ideals
\pages 231-293
\inbook Problems of contemporary approximation theory
\bookinfo
\publ  \vol
\publaddr St. Petersburg, SPb GU
\yr  2004
\endref



\ref \no21 \by Saphar P.\pages 71-100   %$MR 43 \# 878.$
 \paper  Produits tensoriels d'espaces de Banach et classes
d'applications lineaires
 \yr 1970\vol  38
 \jour  Studia Math.
 \endref


\endRefs


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