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%%%%    after Kalinin 1985 Primenen. FA v t. pribl.
%%%%           28.03.2012 9:16:24
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%%%%%%%%%%%%%% poisk voprosov po !!!

\def\nor#1{||{#1}||} %
\def\md#1{|{#1}|}    %
\def\sp#1#2{\(#1,#2\)}           %
\def\ove#1{\overline{#1}}    %
\def\ovs#1#2{\overset{#1}\to{#2}}
     \def\({\left(}       \def\al{\alpha}           \def\lee{\leqslant}
     \def\){\right)}      \def\e{\varepsilon}    \def\gee{\geqslant}
     \def\[{\left[}       \def\la{\lambda}
     \def\]{\right]}      \def\ffi{\varphi}
                          \def\be{\beta}
     %\def\{{\left\{}
                                      \def\ot{\otimes}
     \def\<{\langle}                 \def\wh{\widehat}
     \def\>{\rangle}                 \def\wt{\widetilde}
                 \def\sbs{\subset}
\def\tr{\operatorname{trace}\,}

                   \def\det{\operatorname{det}\,}

  %%%%%%%%%%%%% after 30.01.00 02:44:40 Sat:
\def\Gr{\operatorname{Gr}}
\def\AP{\operatorname{AP}}
\def\BAP{\operatorname{BAP}}
\def\N{\operatorname{N}}
\def\I{\operatorname{I}}
\def\id{\operatorname{id}}
\def\L{\operatorname{L}}
\def\QN{\operatorname{QN}}
\def\J{\operatorname{J}}
\def\R{\operatorname{R}}
\def\reg{\operatorname{reg}}
\def\dual{\operatorname{dual}}
                     \def\sbs{\subset}

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\title{Some more remarks on  Grothendieck-Lidski\v{\i} trace formulas}
%{}
\author{Oleg Reinov}
\address{Department of Mathematics and Mechanics, St. Petersburg State University,
Saint Petersburg, RUSSIA.\newline
\phantom{Ao} Abdus Salam School of Mathematical Sciences, Government College University, Lahore, PAKISTAN.
}
%\address{Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore 54600, PAKISTAN.}
\email{orein51@mail.ru}
%\author{Qaisar Latif}
%\address{Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore 54600, PAKISTAN.}


\thanks{%${ }^\maltese$
The research is supported by the Higher  Education Commission of Pakistan
and by grant 12-01-00216 of RFBR.}

\thanks{%${ }^\maltese$
AMS Subject Classification 2010: 47B06.
}
\thanks{${ }$ Key words: $(s,p)$-nuclear operators, eigenvalue distributions. }




\begin{document}

                          $$ {} $$
\vphantom{} \maketitle


\begin{abstract}
 {\bf Theorem:}\,   {\it
 Let $r\in (0,1], 1\le p\le2,$ $u\in X^*\wh\ot X$ and $u$ admits a representation
 $$
    u=\sum_i \lambda_i x'_i\ot x_i,
    $$
    with $(\la_i)\in l_r,$ $(x'_i)$ bounded and  $(x_i)\in l_{p'}^w(X).$
 If $1/r+1/2-1/p=1,$ then the system $(\mu_k)$ of all eigenvalues  of the corresponding operator $\wt{u}$
 $($written  according to their algebraic multiplicities$)$ is absolutely summable and
 $$
  \tr u=\sum_k \mu_k.
 $$
 }
 One of the main aim of these notes is not only to give a proof of the theorem
 but also to show that it could be obtained by A. Grothendieck in 1955.
 \end{abstract}

\vskip 0.75cm


 In 1955, A. Grothendieck [4] has shown that if the linear operator $T$
in a Banach space  is $2/3$-nuclear then
the trace of $T$ is well defined and is equal to the sum of all  eigenvalues $\{\mu_k(T)\}$
of $T.$  \,
V.B. Lidski\v{\i} [8], in 1959, proved his famous theorem on the coincidence of the trace of the $S_1$-operator
in an (infinite dimensional) Hilbert space
 with its spectral trace $\sum_{k=1}^\infty \mu_k(T).$
 % Any Banach space is a subspace of an $L_\infty(\nu)$-space, as well as any Hilbert space
 % is a (subspace of) $L_2(\nu)$-space. Also, any Banach space is a factor space of an $L_1(\nu)$-space,
 % as well as any Hilbert space is a (factor space of) $L_2(\nu)$-space.
  %We obtain the following generalization of these theorems:
  %{\it
%for $p\in[1,\infty]$ and $s\in (0,1]$ with $1/s=1+|1/2-1/p|,$
%and for every $s$-nuclear operator $T$ in every subspace of any $L_p(\nu)$-space
%the trace of $T$ is well defined and equals the sum of all eigenvalues of $T.$}
 %Note that for $p=2$ one has $s=1,$ and  for $p=\infty$ one has $s=2/3.$
        \vskip 0.08cm

 %in 1955, A. Grothendieck [1] has shown that if the linear operator $T$
%in a Banach subspace of an $L_\infty$-space is $2/3$-nuclear then
%the trace of $T$ is well defined and is equal the sum of all  eigenvalues $\{\mu_k(T)\}$
%of $T.$
%V.B. Lidski\v{\i} [2], in 1959, proved his famous theorem on the coincidence of the trace of an $S_1$-operator
%in $L_2(\nu)$ with its spectral trace $\sum_{k=1}^\infty \mu_k(T).$

In 1970's and in early 1980's, the interest to the trace formulas (and, generally, to the distribution
of eigenvalues of some classes of operators) has been increased (A. Pietsch, H. K{\"o}nig and others).
The trace formula was established for such ideals of operators as $\mathfrak L_1^{app},$\, $\mathfrak{P}_2\circ \mathfrak{P_2},$\,
 $(\mathfrak{P}_2)^{app}_{2,1}$\,  $\mathfrak L_1^{gel},$\,   $\mathfrak L_1^{kol},$\,   $\mathfrak L_1^{weil},$\,
 $\mathfrak L_1^{ent}$\, (see [12], p. 404).
   In the book [10]
   by A. Pietsch, one can find a generalization of Grothendieck-Lidski\v{i} theorem to the case of the quasinormed operator ideal
   $N_{1,1,2}$ of the so called (1,1,2)-nuclear operators (see [10], Th. 27.4.11).        %!!!
  In 1996, M. White [15] has obtained a very general theorem
 on the spectral trace for a wide classes of quasi-normed operator ideals.
            What about concrete Banach spaces, it was shown recently by Oleg Reinov and Qaiser Latif [14] that the
Grothendieck-Lidski\v{i} formula can be "interpolated" between $L_\infty$--$L_2$ (or, between $L_1$--$L_2)$ cases.
More precisely, they have shown that
for $p\in[1,\infty]$ and $s\in (0,1]$ with $1/s=1+|1/2-1/p|,$
and for every $s$-nuclear operator $T$ in every subspace of any $L_p(\nu)$-space
the trace of $T$ is well defined and equals the sum of all eigenvalues of $T.$  The same is true
for quotients of $L_p(\nu)$-spaces.
Note that for $p=2$ one has $s=1,$ and  for $p=\infty$ one has $s=2/3.$

In this note, we  are going to give some more examples of such a kind    %!!!
(see Theorem below).
 Let us mention that the proof of the theorem consists of  a "reconsideration" of some Grothendieck's arguments
 from [4], Chap. II, of proving by him his famous trace formula in the case of 2/3-nuclear operators.
 In the case where $X=H$ is a Hilbert space and $T\in S_1(H)$ (nuclear case; so, $p=2$ in the theorem below),
 our theorem gives the Lidski\v{i}
 formula; in the case where $X$ is any Banach space  and $T$ is 2/3-nuclear (so, $p=+\infty$ in the theorem below),
 we obtain the Grothendieck 2/3-theorem (with an analogues proof!). If $X$ is any and $p=2$ in our theorem,
 we obtain the above mentioned $N_{1.1.2}$-result. We give also (after the proof of the theorem) some new consequences
 and make some remarks on Grothendieck's considerations in Chapter II of his famous work [4].
  Let us note now only that, in particular, A. Pietsch writes (concerning Lidski\v{i}'s 1959 formula) in the book [12], p. 404:       %!!!
  "a remark in [GRO1, Chap. II, p. 13] indicates that
by 1955, Grothendieck was aware of this fact".  We will give a citation from [4], which shows that A. Grothendieck (in 1955)
was aware of a more stronger result than the Lidski\v{i} theorem (but the result was given there without any proof).
Surely, that work by A. Grothendieck was unknown
to V. Lidski\v{i}, so, the famous Lidski\v{i}'s formula is Lidski\v{i}'s  formula forever.

% We will discuss some of the corresponding results and will give  some generalizations
% of Grothen\-dieck-Lidski\v{\i} trace formulas, "interpolating"\, between subspaces of $L_\infty$-
% and $L_2$-spaces.

        \vskip 0.08cm


 %       We were just analyzing some proofs given by A. Grothendieck [Gr1, Ch.2] for his trace formula
 %for $2/3$-nuclear operators and have obtained a generalization  below.

  \vskip 0.33cm


\centerline{\bf \S1. Preliminaries and a theorem}

 \vskip 0.23cm


%{\bf I.}\.
All the terminology and facts (now classical), given here without any explanations, can be found
in [1, 2, 4, 7, 10, 11].                                                                                                                                                                          %!!!
         \vskip 0.02cm
  Let $X,Y$ be Banach spaces. For the Banach dual of $X,$ we use the notation $X^*.$
  If $x\in X$ and $x'\in X^*,$ then we use the notation $\<x',x\>$ for $x'(x).$

 Denote by $X^*\wh\ot Y$
  the completion of the tensor product $X^*\ot Y$  (considered as a linear space
  of all finite rank operators from $X$ to $Y$) with respect to the projective norm
  $$
  ||w||:= \inf \{\(\sum_{k=1}^N ||x'_k||\, ||y_k||\):\ w=\sum_{k=1}^N x'_k\ot y_k\}
  $$
  (see, e.g., [4], [1]).
  For $X=Y,$ the natural linear continuous functional "trace" on $X^*\ot X$ has a unique
 continuous extension to the space $X^*\wh\ot X,$ which we still will denote by "trace".

   Put $N(X,Y):= $ image of $X^*\wh\ot Y$ in the space $L(X,Y)$ of all bounded linear transformations under the canonical factor map
  $X^*\wh\ot Y\to N(X,Y)\sbs L(X,Y).$ We consider the (Grothendieck) space $N(X,Y)$  of all
  nuclear operators from $X$ to $Y$ with the natural norm, induced from $X^*\wh\ot Y.$
 For a tensor element $u\in X^*\wh\ot Y,$ we denote by $\wt{u}$ the corresponding nuclear operator from $X$ to $Y.$

  For $q\in (0,+\infty],$ we denote by $l_q^w(X)$ the space of all weakly $q$-summable sequences
  $(x_i)\sbs X$ (see, e.g., [9], [10]) with a quasi-norm
  $$
   \e_q((x_i)) := \sup \{\(\sum_i |\<x', x_i\>|^q\)^{1/q}:\ x'\in X^*,\, ||x'||\le1\}
  $$
  (in the case where $q=\infty,$ we suppose $(x_i)$ to be just bounded and tending to zero, i.e.,
  $\e_\infty((x_i))=\sup_i ||x_i||$).

  We are going to prove

 {\bf Theorem.}\,   {\it
 Let $r\in (0,1], 1\le p\le2,$ $u\in X^*\wh\ot X$ and $u$ admits a representation
 $$
    u=\sum_i \lambda_i x'_i\ot x_i,
    $$
    with $(\la_i)\in l_r,$ $(x'_i)$ bounded and  $(x_i)\in l_{p'}^w(X).$
 If $1/r+1/2-1/p=1,$ then the system $(\mu_k)$ of all eigenvalues  of the operator $\wt{u}$
 $($written  according to their algebraic multiplicities$)$ is absolutely summable and
 $$
  \tr u=\sum_k \mu_k.
 $$
 }



\vskip 0.1cm

         We obtained this result rather casually,  just analyzing the arguments, given by A. Grothendieck [4, Ch. II]
         for getting his trace formula
 for $2/3$-nuclear operators, and noting that Hadamard's inequality for determinants may be
 improved in some $L_p$ situations (this idea appeared after considerations again of arguments from [14]
 and the facts that the Hilbert spaces are the best Banach spaces, but the
 Banach spaces of type $L_p$ for $p\in(1,\infty)$ are, maybe, worse than an H but better that any $X$
 (or, the same, in a sence, than $L_\infty$)). % and have obtained a generalization  below.

 In the proof of Theorem, we shall use, in particular, the "related operators theorem" [10, p. 375],   %!!!
 namely, in the following situation. If $u$ is as in Theorem then it is easy to see that it admits a factorization
 $$
  u=AB:\ X\to l_p\to X,
 $$
 where $B$ is $s$-nuclear (is generated by "un noyau
de Fredholm de puissance s.\`{e}me sommable dans  $X^*\wh\ot l_p$" in terms of [4]), $A$ maps the unit vector basis of $l_p$
to the sequence $(x_i)$ (which is weakly $p'$-summable). Therefore, the set of all eigenvalues of $u$ is the same
as the set of all (with their algebraic multiplicities) eigenvalues of the operator $BA,$ which maps $l_p$ into $l_p$
(so, results of [6] and [14] may be applied).




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \vskip 0.33cm


\centerline{\bf \S2. Proofs}

 \vskip 0.23cm



%{\bf II.}\,
    Let $u$ be an element of the projective tensor product $X^*\wh\ot X.$
    It can be represented in the form
    $$
    u=\sum_i \lambda_i x'_i\ot x_i,
    $$
where $(\la_i)\in l_1$ and $||x'_i||\le1,$ $||x_i||\le1$ (see [4], [1]).
Recall that the Fredholm determinant $\det (1-zu)$ of $u$ (see [4], [5], [10], [11]) is an entire function
$$
 \det (1-zu)= 1-z\, \tr u+ \dots + (-1)^n z^n\alpha_n(u) +  \dots,
$$
all zeros of which are exactly (according to their multiplicities) the inverses of nonzero eigenvalues $(\mu_k)$
of  the operator $\wt{u},$ associated with the tensor element $u.$
If $u$ has a form  $u=\sum_i \lambda_i x'_i\ot x_i$ as above, the coefficients $\alpha_n(u)$
in the previous formula are defined explicitly by
$$
 \alpha_n= \sum_{i_1<\dots<i_n} \la_{i_1}\dots \la_{i_n}\, \det (\<x'_{i_\alpha}, x_{i_\beta}\>)_{1\le \alpha,\beta\le n}
$$
(see [4, Chap. II, p.13, (5bis)], [5]).

Suppose now, that $u$ has a representation
 $$
    u=\sum_i \lambda_i x'_i\ot x_i,
    $$
    with $(\la_i)\in l_r,$ $\la_i\ge0,$ $r\in(0,1],$ $||x'_i||\le1,$ $(x_i)\in l_{p'}^w(X),$ $ \e_{p'}((x_i))\le1$   (here $1\le p\le2).$
 We have:
 $$
  f(z):= \det (1+zu)= \sum_{n=0}^\infty \alpha_n(u)\, z^n,
 $$
 where $\alpha_n(u)$ are as above; therefore, taking in account that   for every $\alpha=1,\dots, n$
 $$
 \(\sum _{\beta=1}^n |\<x'_{i_\alpha}, x_{i_\beta}\>|^{p'}\)^{1/p'}\le1
 $$
 and thus
 $$
  \(\sum _{\beta=1}^n |\<x'_{i_\alpha}, x_{i_\beta}\>|^{2}\)^{1/2}\le n^{1/p-1/2},
 $$
 by Hadamard's inequality for determinants (see, e.g., [16], 8.7.4 Problems and Exercises, Ex. 9c), or [2, p. 1018]),         %!!!
             %!!!            Vladimir A. Zorich,  Mathematical  Analysis I,  Springer-Verlag Berlin Heidelberg 2004, 597 pages
              %!!!                                 Dunford, N., Schwartz, J. T.   [DUN] Linear operators, vol. I, New York—London 1958
 we get
 $$
   |\alpha_n(u)|\le n^{n(1/p-1/2)}\, \alpha_n(\lambda),
 $$
 where
 $$
  \alpha_n(\la)=\sum_{i_1<\dots<i_n} \la_{i_1}\dots \la_{i_n}.
 $$

Since the function $g(z)=\prod_i (1+\la_i z)$ is of order $\le r$ (see, e.g., [7], p. 30, Th. 3 (Borel))               %!!!
and since its coefficients are exactly $\alpha_n,$   we obtain for these coefficients the estimates, for each $t>r,$
$$
 \alpha_n(\la)\le M_t n^{-n/t}
$$
(see the same book [7], p. 6).
Hence,
$$
 |\alpha_n(u)|\le M_t n^{-n(1/t-1/p+1/2)}=M_t n^{-n/\omega},
$$
where $1/\omega=1/t+1/2-1/p.$
By a classical result of Hadamard (see, e.g., [7], pp. 5-6),
                %!!!   B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographs,  Volume 150
                %!!! American Mathematical Society, Providence, Rhode Island  1996
the function $f(z)$ is of order $\le\omega$ and, therefore, of order $\le \nu,$
where $1/\nu=1/r+1/2-1/p$ (since $t>r$ was arbitrary).

Now, suppose that $\nu=1$ (that is, $1/r+1/2-1/p=1).$
By Hadamard (see [7], p. 26, Th. 1),
$$\det (1-zu)=e^{-az}\, \prod_i (1-z\mu_i)e^{z\mu_i}$$
(recall that $(\mu_k)$ is a sequence of all eigenvalues of $\wt{u},$ counted   according to their algebraic
multiplicities).
On the other hand, as was said above,
$$
 \det (1-zu)= 1-z\, \tr u+ \dots + (-1)^n z^n\alpha_n(u) +  \dots,
$$
and we get (considering the expansion of the entire function $e^{-az}\, \prod_i (1-z\mu_i)e^{z\mu_i}$)         %!!! expan?
that $a=\tr u.$ Therefore,
$$\det (1-zu)=e^{-z\,\tr u}\, \prod_i (1-z\mu_i)e^{z\mu_i}.$$

Now we apply  Theorem 2.6 of [6] or results from [14]
     %!!!    W. B. Johnson, H. K"onig, B. Maurey, and J. R. Retherford, Eigenvalues of p- summing and lp-type operators in Banach spaces, J. Funct. Anal. 32:353-380 ( 1979).
      % Th. 2.6 on Lp Ns kak u nas
 to get that $(\mu_k)\in l_1,$ from which it follows (see, e.g., [7], p. 25-26) that
   $$%\text{1. }\
   \det (1-zu)=e^{-\alpha z}\, \prod_i (1-z\mu_i),\ \, \text{where }\ \alpha=\tr u-\sum_k \mu_k$$
       %where $\alpha=\tr u-\sum_k \mu_k.$
       and
       $$
     %\text{2. }\
     \text{the function }\   \det (1-zu) \ \text{ is of minimal type}
       $$
%Finally, since (by the same Hadamard's theorem) the function $\det (1-zu)$
%is of minimal type
(by the same Hadamard's theorem; see also [7], pp. 25-26 or the second part of the proof of Borel theorem in [7], p. 30).
Whence,  $\alpha=0,$ i.e.
$\tr u=\sum_k \mu_k.$

                          %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \vskip 0.33cm


\centerline{\bf \S3. Corollaries and remarks.}

 \vskip 0.23cm



%\vskip 0.1cm

{\bf Corollary 1.}\it \,
Let $r, p, u$ be as in Theorem. The operator $\wt{u}: X\to X$ is equal to zero iff the tensor element $u$ is zero.
 \rm

\vskip 0.1cm

The same proof as the one of Theorem, with evident changes, gives us
           \vskip 0.1cm

{\bf Corollary 2.}\it \,
 Let $r\in (0,1], 1\le p\le2,$ $u\in X^*\wh\ot X$ and $u$ admits a representation
 $$
    u=\sum_i \lambda_i x'_i\ot x_i,
    $$
    with $(\la_i)\in l_r,$ $(x_i)$ bounded and  $(x'_i)\in l_{p'}^w(X^*).$
 If $1/r+1/2-1/p=1,$ then the system $(\mu_k)$ of all eigenvalues  of the operator $\wt{u}$
 (written  according to their algebraic multiplicities) is absolutely summable and
 $$
  \tr u=\sum_k \mu_k.
 $$

 \rm

\vskip 0.1cm

{\bf Corollary 3.}\it \,
 Let $r, p, u$ be as in the previous corollary. The operator $\wt{u}: X\to X$ is equal to zero
 iff the tensor element $u$ is zero.
 \rm

\vskip 0.1cm

{\it Remark.}\,
For the case where $r=2/3$ and $p=\infty,$ we get  2/3-theorems of A. Grothendieck ([4]; for a simple proof
of the 2/3-theorems, see [13]). For the case $r=1$ and $p=2,$ we get the $N_{1,1,2}$-results of [10, p. 381].                                          %!!!
Also, Corollaries 1 and 3 are valid if we consider the operators $\wt{u}$ from $X$ to $Y,$ for any Banach $X,Y.$   

\vskip 0.1cm

As was said above, in our proof we just used the ideas of A. Grothendieck from [4].
Let us mention that our Theorem could be proved by A. Grothendieck in 1955,
as well as the Lidski\v{i}'s result. Namely, in [4, Ch. II, Remark 4, p. 21], A. Grothendieck writes:                                            %!!!
     \vskip 0.1cm

"{\it Soit $0<p\le1.$ Pour
tout $u$..., soit $\wh{u}$ la suite non ordonnee des valeurs propres
de $u$ ... \, ... ce qui permet facilement, ...,
 de se ramener \`a un r\'esultat plus fin sur les espaces
de Hilbert: Si $H$ est un espace de Hilbert, l'application $u\to\wh{u}$
de $H'\overset{(p)}{\ot} H$ dans\, {\rm (l'espace)}\, $\Sigma_{(p)}$\, {\rm (des suites non ordonn\'ees d'ordre $\le p)$}\,
est continue}."
      \vskip 0.1cm

 Here $\overset{(p)}{\ot}$  denotes the tensor product which corresponds to the space of the $p$-nuclear operators.
 In the case where $p=1,$ we have the class $S_1$ of Schatten and von Neumann.
    Thus, it seems that the $S_1$-trace-formula indeed was known to A. Grothendieck in 1955, but he
 (we can only guess, why) did not pay any more attention to the Hilbert case.
   \vskip 0.1cm

   Concluding the paper (on March 28, 2012), I would like to bring my deep acknowledgments to Alexander Grothendieck
   for his ideas from [4] (which are all in these notes) on the day of his Birth.

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

   \newpage

%%%%%%%%%%       ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


%\bigskip
%\bigskip
%\medskip

\begin{thebibliography}{0}

\bigskip
\medskip


       % Diestel, J./Uhl, J. J. (Junior)
        % [DIU] The theory of vector measures, Providence 1977.
  \bibitem{10} J. Diestel, J.J.Ju. Uhl:
\textit{The theory of vector measures},
Providence (1977).

    % Dunford, N., Schwartz, J. T.   [DUN] Linear operators, vol. I, New York-London 1958

  \bibitem{2} N. Dunford, J.T. Schwartz:
\textit{Linear operators I},
New York-London (1958).

  %     C. Gohberg, M. G. Krein,  Introduction to the theory of linear nonselfadjoint operators
  %     Translations of Mathematical Monographs, Volume 18
   %AMS, Providence, Rhode Island     1969

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 %  Grothendieck, A.
   %La th'eorie de Fredholm, Bull. Soc. Math. France 84 (1956), 319—384
\bibitem{5} A. Grothendieck:
\textit{La th\'eorie de Fredholm},
Bull. Soc. Math. France, \textbf{84} (1956), 319-384.

     %%  W.B. Johnson, H. K"onig, B. Maurey,  J.R. Retherford, Eigenvalues of p-summing and lp-type operators in Banach spaces, J. Funct. Anal. 32:353-380 ( 1979).
\bibitem{6} W.B. Johnson, H. K\"onig, B. Maurey, and J.R. Retherford:
\textit{Eigenvalues of $p$-summing and $l_p$-type operators in Banach spaces},
 J. Funct. Anal., \textbf{32} (1979), 353-380.

    %B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographs,  Volume 150
    %  AMS, Providence, Rhode Island  1996

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 \bibitem{8}
 V.\,B.~Lidski\v{\i}:
  \textit{Nonselfadjoint operators having a trace},
 {Dokl. Akad. Nauk SSSR}, \textbf{125}(1959), 485--487.

  %        Lindenstrauss, J., Tzafriri, L.
  %Classical Banach spaces, vol.1: Sequence spaces, Berlin-Heidelberg-New York
     %1977,
  \bibitem{9} J. Lindenstrauss, L. Tzafriri:
\textit{Classical Banach spaces, vol.1: Sequence spaces},
Berlin-Heidelberg-New York (1977).

    \bibitem{10} A. Pietsch:
\textit{Operator Ideals},
North Holland (1980).

      %A. Pietsch   Eigenvalues and s-Numbers
     %Cambridge studies in advanced mathematics   13
       %     Cambridge University Press           1987

  \bibitem{11} A. Pietsch:
\textit{Eigenvalues and s-Numbers},
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     \bibitem{12}
  A.~Pietsch:
\textit{History of Banach Spaces
and Linear Operators},
Birkh\"auser (2007).

  \bibitem{13} O.I. Reinov:
\textit{A simple proof of two theorems of A. Grothendieck},
Vestn. Leningr. Univ.,  \textbf{7} (1983), 115-116.
%O. I. Reinov, \A simple proof of two theorems of A. Grothendieck, Vestn. Leningr. Univ., 7, 115-116 (1983).

 % Oleg Reinov and Qaisar Latif, Grothendieck-Lidskii theorem for subspaces and
 %factor spaces of Lp-spaces, arXiv:1105.2914v1, electronic preprint, 14 May 2011, 1-4.

\bibitem{14} Oleg Reinov and Qaisar Latif:
\textit{Grothendieck-Lidski\v{\i} theorem for subspaces and
 factor spaces of Lp-spaces},
arXiv: 1105.2914v1, electronic preprint, 14 May  (2011), 1-4.

  \bibitem{15}
  M.C.~White:
\textit{Analytic multivalued functions and spectral trace},
%\textit
{Math. Ann.}, \textbf{304} (1996), 665-683.

  \bibitem{16} V. A. Zorich:
\textit{Mathematical  Analysis I},
Springer-Verlag Berlin Heidelber (2004).

  %Vladimir A. Zorich,  Mathematical  Analysis I,  Springer-Verlag Berlin Heidelberg 2004


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

\end{thebibliography}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \end{document}

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


  %  Grothendieck, A.
   %La th'eorie de Fredholm, Bull. Soc. Math. France 84 (1956), 319—384
\bibitem{4} A. Grothendieck:
\textit{La th\'eorie de Fredholm},
Bull. Soc. Math. France, \textbf{84} (1956), 319-384.

 %     C. Gohberg, M. G. Krein,  Introduction to the theory of linear nonselfadjoint operators
  %     Translations of Mathematical Monographs, Volume 18
   %AMS, Providence, Rhode Island     1969

\bibitem{5} C. Gohberg, M. G. Krein:
\textit{Introduction to the theory of linear nonselfadjoint operators},
Translations of Mathematical Monographs \textbf{18}, AMS, Providence, Rhode Island (1969).


%Journal of Mathematical Sciences, Vol. 115, No. 2, 2003
%Approximation properties $AP_s$ and p-nuclear operators
%(the case 0 <s\le 1)
%O. I. Reinov
%

% Oleg Reinov and Qaisar Latif, Grothendieck-Lidskii theorem for subspaces and
 %factor spaces of Lp-spaces, arXiv:1105.2914v1, electronic preprint, 14 May 2011, 1-4.

\bibitem{6} Oleg Reinov and Qaisar Latif:
\textit{Grothendieck-Lidski\v{\i} theorem for subspaces and
 factor spaces of Lp-spaces},
arXiv: 1105.2914v1, electronic preprint, 14 May  (2011), 1-4.


   \bibitem{7} A. Pietsch:
\textit{Operator Ideals},
North Holland (1980).

   %B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographs,  Volume 150
    %  AMS, Providence, Rhode Island  1996

 \bibitem{8} B. Ya. Levin:
\textit{Lectures on Entire Functions},
Translations of Mathematical Monographs \textbf{150}, AMS, Providence, Rhode Island (1996).


     %%  W.B. Johnson, H. K"onig, B. Maurey,  J.R. Retherford, Eigenvalues of p-summing and lp-type operators in Banach spaces, J. Funct. Anal. 32:353-380 ( 1979).
\bibitem{9} W.B. Johnson, H. K\"onig, B. Maurey, and J.R. Retherford:
\textit{Eigenvalues of $p$-summing and $l_p$-type operators in Banach spaces},
 J. Funct. Anal., \textbf{32} (1979), 353-380.

 \bibitem{10} V. A. Zorich:
\textit{Mathematical  Analysis I},
Springer-Verlag Berlin Heidelber (2004).

  %Vladimir A. Zorich,  Mathematical  Analysis I,  Springer-Verlag Berlin Heidelberg 2004

    % Dunford, N., Schwartz, J. T.   [DUN] Linear operators, vol. I, New York-London 1958

  \bibitem{10} N. Dunford, J.T. Schwartz:
\textit{Linear operators I},
New York-London (1958).

     %A. Pietsch   Eigenvalues and s-Numbers
     %Cambridge studies in advanced mathematics   13
       %     Cambridge University Press           1987

  \bibitem{10} A. Pietsch:
\textit{Eigenvalues and s-Numbers},
Cambridge studies in advanced mathematics  \textbf{13}, Cambridge University Press (1987).

 %        Lindenstrauss, J., Tzafriri, L.
  %Classical Banach spaces, vol.1: Sequence spaces, Berlin-Heidelberg-New York
     %1977,
  \bibitem{10} J. Lindenstrauss, L. Tzafriri:
\textit{Classical Banach spaces, vol.1: Sequence spaces},
Berlin-Heidelberg-New York (1977).

       % Diestel, J./Uhl, J. J. (Junior)
        % [DIU] The theory of vector measures, Providence 1977.
  \bibitem{10} J. Diestel, J.J.Ju. Uhl:
\textit{The theory of vector measures},
Providence (1977).

   \bibitem{3}
  A.~Pietsch:
\textit{History of Banach Spaces
and Linear Operators},
Birkh\"auser (2007).

  \bibitem{4}
  M.C.~White:
\textit{Analytic multivalued functions and spectral trace},
%\textit
{Math. Ann.}, \textbf{304} (1996), 665-683.

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

\end{thebibliography}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \end{document}

  %%%%%%   K Biblio sborka   ???


    [Theorem 8.4 p. 101 in]
   I. C. Gohberg, M. G. Krein,  Introduction to the theory of linear nonselfadjoint operators
       Translations of Mathematical Monographs, Volume 18
AMS, Providence, Rhode Island     1969

    W. B. Johnson, H. K"onig, B. Maurey, and J. R. Retherford, Eigenvalues of p- summing and lp-type operators in Banach spaces, J. Funct. Anal. 32:353-380 ( 1979).

 B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographs,  Volume 150
      American Mathematical Society, Providence, Rhode Island  1996

  Vladimir A. Zorich,  Mathematical  Analysis I,  Springer-Verlag Berlin Heidelberg 2004, 597 pages

 Dunford, N., Schwartz, J. T.   [DUN] Linear operators, vol. I, New York—London 1958

  A. Pietsch   Eigenvalues and s-Numbers
     Cambridge studies in advanced mathematics   13
            Cambridge University Press           1987

 Oleg Reinov and Qaisar Latif, Grothendieck-Lidskii theorem for subspaces and
factor spaces of Lp-spaces, arXiv:1105.2914v1, electronic preprint, 14 May 2011, 1-4.

 Lindenstrauss, J., Tzafriri, L.
  Classical Banach spaces, vol.1: Sequence spaces, Berlin—Heidelberg—New York
1977, vol. II: Function spaces, Berlin—Heidelberg —New York 1979.

 Diestel, J./Uhl, J. J. (Junior)
[DIU] The theory of vector measures, Providence 1977. 

 Grothendieck, A.
  La th'eorie de Fredholm, Bull. Soc. Math. France 84 (1956), 319—384
  












  %%%%%%%%%%%%%%%%%%%%%%%%%%    OLDOLDOLDOLD %%%%%%%%%%%%%%%%%%%%%%%
{\bf Definition.}\
We say that $T$ {\it possesses the property $AP_s$}\,
(written down as "$T\in AP_s$") if  for every $X$ and any tensor element $z\in X^*\wh\ot_s Y$
the operator $T\circ z: X\to W$ is zero iff the corresponding tensor $({\mathbf1}\ot T)(z)$
is zero as an element of the space $X^*\wh\ot W.$   If $Y=W$ and $T$ is the identity map, we write just
$Y\in AP_s$ (the approximation property of order $s).$
\vskip 0.1cm

This is equivalent to the fact that if $z\in X^*\wh\ot_s Y$  then it follows from
$$
 \tr (\mathbf1\ot T)(z)\circ R=0, \ \ \forall\, R\in W^*\ot X
$$
that $\tr U\circ (\mathbf1\ot T)(z)=0$ for every  $U\in L(W,X^{**}).$
There is a simple characterization of the condition $T\in AP_s$
in terms of the approximation of $T$ on some sequences of the space $Y,$
but we omit it now, till the next time.
We need here only one example which is crucial for our note (other
examples, as well as more general applications will appear elsewhere).
     \vskip 0.1cm

     {\bf Example.}\
     Let $s\in (0,1],$ $p\in [1,\infty]$ and $1/s=1+|1/p-1/2|.$
     Any subspace as well as any factor space
      of any $L_p$-space have the property $AP_s$
     (this means that, for that space $Y,$  $\id_Y\in AP_s).$
     Thus, in the case of such a space $Y,$ we have the quasi-Banach equality
     $X^*\wh\ot_s Y=N_s(X,Y),$ whichever the space $X$ was.

       \vskip 0.1cm

{\bf Lemma.}\
Let $s\in (0,1],$ $p\in [1,\infty]$ and $1/s=1+|1/2-1/p|.$
Then the system of all eigenvalues (with their algebraic multiplicities)
of any operator $T\in N_s(Y,Y),$ acting in any subspace $Y$ of any
$L_p$-space, belongs to the space $l_1.$ The same is true for the factor spaces
of $L_p$-spaces.

           \vskip 0.1cm

           {\bf Corrolary.}\
           If    $s\in (0,1],$ $p\in [1,\infty]$ with $1/s=1+|1/2-1/p|$
           then the quasi-normed ideals $\Phi_{p,s}$ and $\Phi_{s,p}$
           are of (spectral) type $l_1.$
   \vskip 0.1cm

  {\bf Theorem.}\
Let $Y$ be a subspace or a factor space of  an $L_p$-space,
$1\le p\le \infty.$ If $T\in N_s(Y,Y),$\,
$1/s=1+|1/2-1/p|,$   \,
then

1.\, the (nuclear) trace  of $T$ is well defined,

2.\, $\sum_{n=1}^\infty |\la_n(T)|<\infty,$ where
$\{\la_n(T)\}$ is the system of all eigenvalues of the operator $T$
(written in according to their algebraic multiplicities)

and
$$
 \tr T= \sum_{n=1}^\infty \la_n(T).
$$

\vskip 0.3cm
%\newpage

\centerline{\bf \S2. Proofs}

 \vskip 0.2cm

{\it Proof}\ of Lemma.    \
Let $Y$ be a subspace or a factor space of an $L_p$-space
and $T\in N_s(Y,Y)$ with an s-nuclear representation
$$
 T=\sum_{k=1}^\infty \mu_k y'_k\ot y_k,
$$
where $||y'_k||, ||y_k||=1$ and $\mu_k\ge 0,$  $\sum_{k=1}^\infty \mu_k^s<\infty.$
%Let-... omited
The operator $T$ can be factored in the following way:
$$
 T: Y\overset{A}\longrightarrow l_\infty \overset{\Delta_{1-s}}\longrightarrow l_r
 \overset{j}\hookrightarrow c_0 \overset{\Delta_s}\longrightarrow l_1\overset{B}\longrightarrow Y,
$$
where $A$ and $B$ are linear bounded, $j$ is the natural injection, $\Delta_s\sim(\mu_k^s)_k$ and
$\Delta_{1-s}\sim(\mu_k^{1-s})$ are the natural diagonal operators from $c_0$ into $l_1$ and
from $l_\infty$ into $l_r,$ respectively. Here, $r$ is defined via the conditions
 $1/s=1+|1/p-1/2|$ and $\sum_k \mu_k^s<\infty:$
 we have to have $\sum_k \mu_k^{(1-s)r}<\infty,$ for which $(1-s)r=s$ is good. Therefore, put
 $1/r=1/s-1,$ or $1/r=|1/p-1/2|.$

        from now onward in the proof we assume (surely, without loss of generality) that
        $p\ge2.$
Then $1/r=1/2-1/p$ and $r(1-s)=s.$
Note that if $s=1$ then $r=\infty, p=2$ and $j\Delta_{1-s}\equiv j;$
and if $s=2/3$ then  $r=2, p=\infty$ and $\Delta_{1-s}\sim (\mu_k^{1/3})_k\in l_2.$

Now, let us factorize the diagonal $\Delta_s$ as
$\Delta_s=\Delta_2\Delta_1:\, c_0 \overset{\Delta_{1}}\longrightarrow l_2  \overset{\Delta_{2}}\longrightarrow l_1$
in such a (clear) way that diagonals $\Delta_1$ is in $\Pi_2$ and $\Delta^*_2$ is in $\Pi_2$ too, respectively.

{\it Case }\, (i).\ $Y$ is a subspace of an $L_p$-space.
Denoting by $l: Y\hookrightarrow L_p$ an isomorphic embedding of $Y$ into a corresponding $L_p=L_p(\nu),$
we obtain that the map
$\Delta^*_2B^*l^*: \, L_{p'}\overset{l^*}\longrightarrow Y^*\overset{B^*}\longrightarrow l_\infty \overset{\Delta^*_{2}}\longrightarrow l_2$
is of type $\Pi_2,$    so is in $\Pi_p.$
Thus its preadjoint $lB\Delta_2:\, l_2\overset{\Delta_2}\longrightarrow l_1\overset{B}\longrightarrow Y \overset{l}\longrightarrow L_p$
is order bounded and, therefore, $p$-absolutely summing.

{\it Case }\, (ii).\ $Y$ is a factor space of an $L_p$-space.
Denoting by $q: L_p\to Y$ a factor map from a corresponding $L_p=L_p(\nu)$ onto $Y$ and
taking a lifting $Q: l_1\to L_p$ for $B$ with $B=qQ,$
we obtain that the map
$\Delta^*_2Q^*: \, L_{p'}\overset{Q*}\longrightarrow l_\infty \overset{\Delta^*_{2}}\longrightarrow l_2$
is of type $\Pi_2,$    so is in $\Pi_p.$
Thus its pre-adjoint $Q\Delta_2:\, l_2\overset{\Delta_2}\longrightarrow l_1\overset{Q}\longrightarrow L_p$
is order bounded and, therefore, $p$-absolutely summing.
Hence, $B\Delta_2: \, l_2\overset{\Delta_2}\longrightarrow l_1\overset{Q}\longrightarrow L_p\overset{q}\longrightarrow Y$
is also $p$-absolutely summing.

It folows from all that's said that in all the cases our operator $T:Y\to Y$ can be written as a composition:
 $$
  T=U_1U_2U_3\ \text{ with } \ U_3\in\Pi_r, U_2\in \Pi_2, U_1\in \Pi_p,
 $$
all the exponents being not less than 2.
 Now, $1/r+1/2+1/p=(1/2-1/p)+1/2+1/p=1.$
 %Therefore, Lemma is proved.
  \QQ

    \vskip 0.1cm

 {\it Proof}\ of the statement of Example.
  It follows from:

 $(\alpha)$\ every finite dimensional subspace $E$ of any factor space of any $L_p$-space
is $c_p\,(\dim{E})^{|1/2-1/p|}$-complemented.

\noindent
 For more general statements on $AP_s$ and their proofs,
we refer to [4] and [5]; see also an old paper of O.I. Reinov [3] for the idea to apply the projections
in the questions which are under consideration in this note.     \QQ

          \vskip 0.1cm

 {\it Proof}\ of Corrolary.
 Apply Lemma.        \QQ

      \vskip 0.1cm

 {\it Proof}\ of Theorem.
 Apply Lemma, Example, Corrolary and the main  result of M.C. White [6].

     \vskip 0.1cm

 {\it Remark}:\,
 Since finite rank operators are dense in $N_s,$
 Theorem can be proved without referring to the paper of M.C. White;
  but this would take a little bit longer explanations.


   \newpage

%%%%%%%%%%       ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


%\bigskip
%\bigskip
%\medskip

\begin{thebibliography}{0}

\bigskip
\medskip

\bibitem{1}
 A.~Grothendieck:
 \textit{Produits tensoriels topologiques et \'espaces nucl\'eaires},
{Mem. Amer. Math. Soc.}, \textbf{16}(1955).

 \bibitem{2}
 V.\,B.~Lidski\v{\i}:
  \textit{Nonselfadjoint operators having a trace},
 {Dokl. Akad. Nauk SSSR}, \textbf{125}(1959), 485--487.

 \bibitem{3} O.I. Reinov:
\textit{A simple proof of two theorems of A. Grothendieck},
Vestn. Leningr. Univ.  7 (1983), 115-116.
%O. I. Reinov, \A simple proof of two theorems of A. Grothendieck, Vestn. Leningr. Univ., 7, 115-116 (1983).

\bibitem{4} O.I. Reinov:
\textit{Disappearance of tensor elements in the scale of p-nuclear operators},
Theory of operators and theory of functions (LGU) 1(1983), 145-165.


\bibitem{5} O.I. Reinov:
\textit{Approximation properties $AP_s$ and p-nuclear operators
(the case $0 <s\le 1)$},
Journal of Mathematical Sciences \textbf{115}, No. 3 (2003), 2243-2250.


%Journal of Mathematical Sciences, Vol. 115, No. 2, 2003
%Approximation properties $AP_s$ and p-nuclear operators
%(the case 0 <s\le 1)
%O. I. Reinov
%

%[35] M.C. White, Analytic multivalued functions and spectral trace, Math. Ann. 304 (1996), 665-683.

\bibitem{6} M.C. White:
\textit{Analytic multivalued functions and spectral trace},
Math. Ann. 304 (1996), 665-683.


   \bibitem{7} A. Pietsch:
\textit{Operator Ideals},
North Holland (1980).

 \bibitem{8} P. Wojtaszczyk:
\textit{Banach Spaces for Analysts},
Cambridge Univ. Press (1991).

\bibitem{9} Hermann K\"onig:
\textit{Eigenvalues of Operators and Applications},
 Handbook of the geometry of Banach spaces, vol. 1, Chapter 22 (2001), 941-974.

 \bibitem{10} A. Pietsch:
\textit{History of Banach Spaces
and Linear Operators},
Birkh\"auser (2007).


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

\end{thebibliography}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

%%%%%%%%
%%%%%%%%
%%%%%%%%
  [30] A. Pietsch, Operator Ideals, North Holland (1980).

 Handbook of the geometry of banach spaces, vol. 1           CHAPTER 22
  Eigenvalues of Operators and Applications        941-974
         Hermann K\"onig
© 2001 Elsevier

[192] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ. Press (1991).

Albrecht Pietsch
 History of Banach Spaces
and Linear Operators
  Birkh\"auser                  2007 Birkha?user Boston
6%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{1} Sinha D.P., Karn A.K.:
\textit{Compact operators which factor through subspaces of $l_p$},
Math. Nachr.  281(2008), 412-423.

\bibitem{2} O.I. Reinov:
\textit{Approximation properties of order p and the existence of
 non-p-nuclear operators with p-nuclear second adjoints},
Math. Nachr.  109(1982), 125-134.



\bibitem{3} O.I. Reinov:
\textit{Approximation of operators in Banach spaces},
Application of functional analysis in the approximation theory (KGU, Kalinin)
  (1985), 128-142.

\bibitem{4} O.I. Reinov:
\textit{Disappearance of tensor elements in the scale of p-nuclear operators},
Theory of operators and theory of functions (LGU) 1(1983), 145-165.


\bibitem{5} O. Reinov: \textit{Approximation properties of order p and the existence of
  non-p-nuclear operators with p-nuclear second adjoints},
  Doklady AN SSSR, 256(1981), 43--47.



\bibitem{6} A. Pietsch:
Operator ideals, North-Holland, Deutscher Verlag der Wiss., Berlin,
1978.

\bibitem{7} P. Saphar:
\textit{Produits tensoriels d'espaces de Banach et classes
d'applications lineaires},
Studia Math. 38(1970), 71--100.


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\end{thebibliography}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

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




\ref \no133   \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 \no134 \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

%%%%%%%%

    \newpage
 %\comment

where
$ (jU\Phi_K)^{**}\Psi:
    C(K)\ovs{\Psi}\longrightarrow (Y_K)^{**}
         \ovs{\Phi_K^{**}}\longrightarrow Y
       \ovs{U}\longrightarrow X\ovs{j}\longrightarrow C(K).$
% identify X with its image in  X**; come through with a compact act!!!!
Since
$ \pi_p(A_n- jU\Phi_K)\to 0,$
then
$ \pi_p\(A^{**}_n - (jU\Phi_K)^{**}\)\to 0.$
Moreover, if
$A:= A_n=\sum_1^N w_m\ot f_m\in (Y_K)^*\ot C(K),$
then
$$ \tr A_n^{**}\Psi= \sum_m \< \Psi^*w_m, f_m\> =
   \sum_m \< w_m, \Psi f_m\>= \tr \Psi A.
$$

\begin{multline}
  \tr U\circ z= \tr \( \sum j^*(\mu_n)\ot Uy_n\)=
   \sum \< j^*(\mu_n), Uy_n\> =\\ =
\sum \< \mu_n, jUy_n\>=
\tr jU\Phi_K^{**}\Psi=\tr (jU\Phi_K)^{**}\Psi, \label(3)
\end{multline}







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

\vskip 0.1cm

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

\vskip 0.1cm

{\bf Proposition 2} [3]. \it
If the canonical mapping
$ j: X^*\wh\ot_{p'} Y\to \N_{p'}(X,Y)$
is one-to-one then
$ \Pi_p(Y,X)= \ove{Y^*\ot X}^{\,\tau_p}.$
\vskip 0.1cm

\rm
{\it Proof}
If the map


Therefore,
$ \Pi_p(Y,X)= \ove{Y^*\ot X}^{\,\tau_p}.$
\vskip 0.1cm


For a reflexive space
$ X,$
the dual space to
$ X^*\wh\ot_{p'} Y$
is equal to
$ \Pi_p(Y,X).$
Consequently, it follows from the last two statements

\vskip 0.1cm

{\bf Corollary 2.}\it \, For a reflexive space
$ 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(Y,X)$
in the topology
$ \tau_p$\ of $ \pi_p$-compact convergence. \rm

\vskip 0.1cm




\vskip 0.1cm
 {\bf Theorem.}\it \, Let $1\le q\neq2 \le\infty.$ Then there is a $($reflexive$)$ Banach space
that fails the approximation property of type $q$
of $[1].$

\rm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

       % \endcomment
