%%%%%%%%%% %%%%% %%%% APsAPPL II 18.05.2017 8:24:10 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% poisk voprosov po !!! % \def\ove#1{\overline{#1}} % \def\ovs#1#2{\overset{#1}\to{#2}} \def\({\big(} \def\al{\alpha} \def\lee{\leqslant} \def\){\big)} \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\id{\operatorname{id}\,} \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\dim{\operatorname{dim}} \def\dual{\operatorname{dual}} \def\sbs{\subset} \def\co{\operatorname{co}\,} \def\span{\operatorname{span}} \def\op{\operatorname} %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% poisk voprosov po !!! %%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt,oneside]{amsproc} \usepackage[T2A]{fontenc} %\usepackage[cp1251]{inputenc} \usepackage[english]{babel} %\usepackage[russian]{babel} \usepackage{amssymb} \usepackage{amsthm} \usepackage{verbatim} \textheight=22cm \textwidth=15cm \oddsidemargin=5mm \topmargin=-5mm \title[Approximation properties associated with operator ideals] {Approximation properties associated with quasi-normed operator ideals of $(r,p,q)$-nuclear operators} %{} \author{Oleg Reinov} \address{ St. Petersburg State University, Saint Petersburg, RUSSIA.} \email{orein51@mail.ru} \thanks{%${ }^\maltese$ AMS Subject Classification 2010: 46B28 Spaces of operators; tensor products; approximation properties. } \thanks{${ }$ Key words: nuclear operator; tensor product; approximation property; eigenvalue.} \begin{document} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} We consider quasi-normed tensor products lying between Laprest\'e tensor products and spaces of $(r,p,q)$-nuclear operators. We define and investigate the corresponding approximation properties for Banach spaces. An intermediate aim is to answer a question of Sten Kaijser. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip %%%%%%%%%%%%%%%%%%%%%% BEGINNING % {\bf Introduction.}\, %!!! Begin text with this %%% \begin{comment} In 1955, A. Grothendieck \cite{Gr} has introduced the notion of the %!!!ref approximation property (the AP) and has shown that {\it there exists a Banach space without the approximation property iff there exists an operator $U: c_0\to c_0$ such that the (nuclear) trace of $U$ equals 1 but $U^2=0$ identically}\, \cite[Chap. I, "Proposition" 37. $(a')\Leftrightarrow (f''),$ pp. 170-171]{Gr}. %!!!ref Let us recall a proof of the "only if" part of this result. As is known \cite{Gr}, the AP for $X$ is equivalent to the injectivity %!!!ref of the canonical map from the projective tensor product $X^*\wh\ot X$ into the space $L(X)$ of all linear operators in $X.$ Suppose $X$ does not have the AP and let $z$ be an element of the projective tensor product $X^*\wh\ot X$ such that the associated operator $\wt z$ is identically zero. Take a representation $z=\sum \mu_j x'_j\ot x_j$ in $X^*\wh\ot X$ with $1\ge ||x'_j||\to 0,\, ||x_j||=1,$ $(\mu_j)\in l_1$ and $\tr z=1$ (note that $||z||_{\land} \le \sum \mu_j ||x'_j||).$ Consider a diagram $$ X\overset{A}\to c_0 \overset{\Delta}\to l_1 \overset{B}\to X \overset{A}\to c_0 \overset{\Delta}\to l_1 \overset{B}\to X \overset{A}\to c_0, $$ where $Ax= (\)\in c_0,$ $\Delta (\al_j)= (\mu_j \al_j),$ $B(\al_j)= \sum_j \beta_j x_j$ and $B\Delta A= \wt z=0.$ Putting $T:= AB\Delta$ we get a nuclear operator $T$ with $T^2=0$ and $T(\al_j)= \sum_k \mu_k \al_k ABe_k,$ where $(e_k)$ is a sequence of unit vectors i $c_0$ (and also in $l_1).$ Thus, $T= \sum \mu_k e_k\ot Ax_k \in l_1\wh\ot c_0$ and $\tr T= \sum \mu_j \\>= \sum \mu_k \= \tr z =1.$ Note that $||T||_\land\le \sum \mu_j$ while $||z||_\land\le \sum \mu_j) ||x'_j||$ (where $x'_j\to0).$ \end{comment} %%% %!!! locref $$...$$ This is, essentially, a continuation of the author's paper \cite{Trend} %!!!ref and the first part of the work on the approximation properties connected with the quasi-normed ideals of so-called $(r,p,q)$-nuclear operators. \bigskip %% \centerline{\bf 0. Notation, preliminaries}\, \bigskip Throughout, we denote by $X,Y,\dots, G,F,W\dots$ Banach spaces over a field $\mathbb K$ (which is either $\mathbb R$ or $\mathbb C);$ $X^*, Y^*,\dots$ are Banach dual to $X, Y, \dots.$ By $x, y,\dots, x',\dots$ (maybe with indices) we denote elements of $X, Y, \dots, Y^*\dots$ respectively. $\pi_Y: Y\to Y^{**}$ is a natural isometric imbedding. Notations $l_p,$ $l_p^n$ $(0$ does not depend on representation of $z$ in $X^*\ot X.$ $L(X,Y)$ is a Banach space of all linear continuous mappings ("operators") from $X$ to $Y$ equipped with the usual operator norm. %We have: %$X\ot Y\subset F(X^*, Y)\subset L(X^*,Y);$ a completion of the algebraic %tensor product $X\ot Y$ with respect to the operator norm is denoted by %$X\wh\ot Y$ ("") If $A\in L(X,W),$ $B\in L(Y,G)$ and $z\in X\ot Y,$ then a linear map $A\ot B: X\ot Y\to W\ot G$ is defined by $A\ot B((x\ot y):= Ax\ot By$ (and then extended by linearity). Since $\wt{A\ot B(z)}= B\wt z A^*$ for $z\in X\ot Y,$ we will use notation $B\circ z \circ A^*\in W\ot G$ for $A\ot B(z).$ In the case where $X$ is a dual space, say $F^*,$ and $T\in L(W,F)$ (so, $A=T^*: F^*\to W^*),$ one considers a composition $B\wt z T;$ in this case $T^*\ot B$ maps $F^*\ot Y$ into $W^*\ot Y$ and we use notation $B\circ z\circ T$ for $T^*\ot B (z).$ If $\nu$ is a tensor quasi-norm (see \cite[0.5]{166}), %on $X\ot Y$ and $W\ot G$, %!!!ref then $\nu(A\ot B (z))\le ||A||\, ||B||\, \nu(z)$ and we can extend the map $A\ot B$ to the completions of the tensor products with respect to the quasi-norm $\nu,$ having the same inequality. The natural map $(X\ot Y, \nu)\to L(X^*, Y)$ is continuous and can be extended to the completion $\wh{X\ot_\nu Y};$ for a tensor element $z\in \wh{X\ot_\nu Y},$ we still denote by $\wt z$ the corresponding operator. The natural mapping $\wh{X\ot_\nu Y}\to L(X^*, Y)$ need not to be injective; {\it if it is injective for a fixed $Y$ and for all $X,$ then we say that $Y$ has the $\nu$-approximation property}. A projective tensor product $X\wh\ot Y$ of Banach spaces $X$ and $Y$ is defined as a completion of $X\ot Y$ with respect to the norm $||\cdot||_{\land}:$ if $z\in X\ot Y,$ then $$ ||z||_\land:= \inf \sum_{k=1}^n ||x_k||\, ||y_k||, $$ where infimum is taken over all representation of $z$ as $\sum_{k=1}^n x_k\ot y_k.$ We can try to consider $X\wh\ot Y$ also as operators $X^*\to Y$ or $Y^*\to X,$ but this correspondence is, in general, not one-to-one. Note that $X\wh\ot Y= Y\wh\ot X$ in a sence. If $z\in X\wh\ot Y, \e>0,$ then one can represent $z$ as $z=\sum_{k=1}^\infty x_k\ot y_k$ with $\sum_{k=1}^\infty ||x_k||\, ||y_k||<||z||_\land+\e.$ For $z\in X^*\wh\ot X$ with a "projective representation" $z=\sum_{k=1}^\infty x'_k\ot x_k,$ trace of $z, \tr z:=z=\sum_{k=1}^\infty \,$ does not depend of representation of $z.$ The Banach dual $(X\wh\ot Y)^*= L(Y,X^*)$ by $\=\tr T\circ z.$ Some more notations: If $\frak A$ is an operator ideal then $\frak A^{reg}(X,Y):= \{T\in L(X,Y):\ \pi_YT\in \frak A(X,Y^{**})\},$ $\frak A^{dual}(X,Y):= \{T\in L(X,Y):\ T^*\in \frak A(Y^*,X^*)\}.$ Finally, $$l_p(X):=\{(x_i)\subset X:\ ||(x_i)||_p:=\(\sum ||x_i||^p)^{1/p}<\infty\},$$ $$l_\infty(X):=\{(x_i)\subset X:\ ||(x_i)||_\infty:=\sup_i ||x_i||<\infty\},$$ $$l^w_p(X):=\{(x_i)\subset X:\ ||(x_i)||_{w,p}:= \sup_{||x'||\le1}\(\sum |\|^p)^{1/p}<\infty\},$$ $$l^w_\infty(X):=\{(x_i)\subset X:\ ||(x_i)||_{w,\infty}:=\sup_i ||x_i||<\infty\}.$$ Note that if $p\le q,$ then $||\cdot||_q\le ||\cdot||_p$ and $||\cdot||_{w,q}\le ||\cdot||_{w,p}.$ If $0= \$ \cite[pp. 56-57]{166}. %!!!ref Recall that, by definition, an operator $T: X\to F$ is absolutely $(\infty,p,q)$-summing if for any finite sequences $(x_k)$ and $(f'_k)$ (from $X$ and $F^*$ respectively) one has $$ \sup_k |\|\le C\, ||(x_k)||_{w,p} ||(f'_k)||_{w,q}. $$ With a norm $\pi_{\infty,p,q}(T):= \inf C,$ the space $\Pi_{\infty,p,q}(X, F)$ is a Banach space and in duality above (for $F=Y^*)$ $\pi_{\infty,p,q}(T)= ||\tau||$ (on the right is the norm of the functional $\tau$ in $(\wh{X\ot_{r,p,q} Y})^*$). Futhermore, taking a sequence in $X\times F^*,$ consisting of one nonzero element $(x,f'),$ we obtain: If $T\in \Pi_{\infty,p,q}(X, F),$ then $|\|\le \pi_{\infty,p,q}(T)\, ||x||\, ||f'||;$ thus, $||T||\le \pi_{\infty,p,q}(T).$ On the other hand, if $T\in L(X,F),$ then for any finite sequences $(x_k)$ and $(f'_k)$ we have: $$\sup_k |\|\le ||T||\, \sup_k ||x_k||\, \sup_i ||f'_i||\le ||T||\, ||(x_k)||_{w,p}\, ||(f'_k)||_{w,q}.$$ Therefore, $\Pi_{\infty,p,q}(X, F)= L(X, F).$ I do not know whether the dual space $\Pi_{\infty,p,q}(X, Y^*)$ separates points of $\wh{X\ot_{r,p,q} Y}.$ If so, then the natural map $\wh{X\ot_{r,p,q} Y}\to X\wh\ot Y$ is one-to-one. As a matter of fact, it follows from the above considerations, that {\it the space $\Pi_{\infty,p,q}(X, Y^*)$ separates points of $\wh{X\ot_{r,p,q} Y}$ iff the natural map $j_{r,p,q}:\, \wh{X\ot_{r,p,q} Y}\to X\wh\ot Y$ is one-to-one.} \smallskip {\bf Definition 1.1.}\, We define a tensor product $X\wh\ot_{r,p,q} Y$ as a linear subspace of the projective tensor product $X\wh\ot Y,$ consisting of all tensor elements $z,$ which admit representations of type $$ z=\sum_{k=1}^\infty \al_k x_k\ot y_k,\ (\al_k)\in l_r,\, (x_k)\in l_{w,p},\, (y_k)\in l_{w,q} $$ and equipped with the quasi-norm $||z||_{\land\!; r,p,q}:= \inf ||(\al_k)||_r\, ||(x_k)||_{w,p}\, ||(y_k)||_{w,q},$ where the infimum is taken over all representations of $z$ in the above form. \smallskip Note that this tensor product is $\beta$-normed (see \cite{166, PiOP}). %!!! ref Check!!! \smallskip {\it Remark 1.1}.\, We can define $X\wh\ot_{r,p,q} Y$ also as a quotient of the space $\wh{X\ot_{r,p,q} Y}$ by the kernel of the map $j_{r,p,q}$ (i.e. by the annihilator $L(X,Y^*)_{\perp}$ of $L(X,Y^*)$ in the space $\wh{X\ot_{r,p,q} Y}).$ Therefore: (i)\, The tensor product $X\wh\ot_{r,p,q} Y$ is complete, i.e. a quasi-Banach space. This, with the injectivity of the natural map $X\wh\ot_{r,p,q} Y\to X\wh\ot Y$ answers a corresponding question of Sten Kaijser ("Why the last map is one-to-one for the "completion" $X\wh\ot_{r,p,q} Y$?"). (ii)\, If the dual of $\wh{X\ot_{r,p,q} Y}$ separates points of this space, then we can write $\wh{X\ot_{r,p,q} Y}= X\wh\ot_{r,p,q} Y.$ In this case "finite nuclear" quasi-norm $\mu_{r,p,q}$ coincides with the tensor quasi-norm $||z||_{\land\!; r,p,q}$ (compare with \cite[18.1.10.]{PiOP}). %!!! ref (iii)\, The dual space to $X\wh\ot_{r,p,q} Y$ is still $\Pi_{\infty,p,q}(X,Y^*)$ of absolutely $(\infty,p,q)$-summing operators from $X$ to $Y^*$ with its natural quasi-norm. \medskip It follows from Lemma 1.1 %!!! L.1--> Prop. (or, if one wishes, can be proved by the same method) \medskip {\bf Proposition 1.1}\, Let 1)\, $00$ and choose a representation for $z,$ $z=\sum_{k=1}^\infty \al_k\, x_k\ot y_k,$ with $||(\al_k)||_r\, ||(x_k)||_{w,p}\, ||(y_k)||_{w,q}\le ||z||_{\land\!; r,p,q}\, (1+ \delta).$ Let $E:= R(Y)\subset Y,$ $M:= \dim E$ and $\e=\e(M)\in(0,\delta]$ (to be chosen later). Then $\wt z= \sum_{k=1}^\infty \al_k\, \ \ffi_k,$ where $\ffi_k=Ry_k\in E.$ Let $N$ be such that $||(\al_k)_N^\infty||_r\, ||(x_k)_N^\infty||_{w,p}\, ||\ffi_N^\infty||_{w,q}\le \e\, ||z||^\beta_{\land\!; r,p,q}.$ Now, since $E$ is finite dimensional, $\id_E$ admit a representation in $\wh{E^*\ot_{r,p,q} E}$ which give us an estimation from above for $\mu_{r,p,q}(\id_E)$ by a constant $C=C(M)$ depending only on $M.$ %!!! \mu -> to ||.||_.. and around Indeed, take an isomorphism $A: E\to l_2^M$ with $||A||=1, ||A^{-1}||\le \sqrt M$ (see e.g. \cite[Corollary 3.9]{Pis}). %!!! ref %Pisier, Gilles, % (Cambridge tracts in mathematics; 94), Cambridge University Press 1989, The volume of convex bodies and Banach space geometry Since $\id_{l_2^M}=\sum_{k=1}^M e'_k\ot e_k,$ \, $\mu_{r,p,q}(\id_{l_2^M})\le M^{1/\beta}.$ Therefore, $\id_E=A^{-1} \id_{l_2^M} A = \sum_{k=1}^M A^*e'_k\ot A^{-1}e_k$ and $\mu_{r,p,q}(\id_E)\le M^{1/\beta+ 1/2}.$ So, for any $v\in X\ot E,$ considering $\id_E\circ v$ $(= \id_E \wt v)$ we obtain an inequality $\mu_{r,p,q}(\id_E\circ v)\le M^{1/\beta+ 1/2}\, ||\wt v||.$ Since $||\wt v||\le ||v||_{\land\!; r,p,q}$ we get $$ \mu_{r,p,q}(\id_E\circ v)\le C(M)\, ||v||_{\land\!; r,p,q}. $$ Hence, for our $z$ we get $$ \mu_{r,p,q}^\beta(z)\le \mu_{r,p,q}^\beta\(\sum_{k=1}^N \al_k\, x_k\ot \ffi_k\) + \mu_{r,p,q}^\beta\(\sum_{k=N+1}^\infty \al_k\, x_k\ot \ffi_k\) \le $$ $$ %(1+\delta)^\beta\, ||R||^\beta\, \mu_{r,p,q}^\beta\(\sum_{k=1}^N \al_k\, x_k\ot y_k\) + % \e^\beta C(M)^\beta\, ||\sum_{k=N+1}^\infty \al_k\, x_k\ot \ffi_k||^\beta_{\land\!; r,p,q}\le $$ $$ % (1+\delta)^\beta\, ||R||^\beta\, (||(\al_k)||_r\, ||(x_k)||_{w,p}\, ||(y_k)||_{w,q})^\beta + C(M)^\beta\, (||(\al_k)_N^\infty||_r\, ||(x_k)_N^\infty||_{w,p}\, ||\ffi_N^\infty||_{w,q})^\beta \le $$ $$ (1+\delta)^\beta\, ||R||^\beta\, ||z||^\beta_{\land\!; r,p,q}+ \e^\beta C(M)^\beta\, ||z||^\beta_{\land\!; r,p,q}. $$ Taking $\e<\delta$ to have $\e^\beta C(M)^\beta<\delta,$ we obtain $\mu_{r,p,q}^\beta(z) \le [(1+ \delta)^\beta\, (C +1)^\beta +\delta]\, ||z||^\beta_{\land\!; r,p,q}.$ This inequality means, in particular, that the quasi-norms $\mu_{r,p,q}$ and $||\cdot||_{\land\!; r,p,q}$ are equivalent on the space $X\ot Y,$ thus giving the same completions. Since the space $X\wh\ot_{r,p,q} Y$ is complete (Remark (i) above), we obtain that the natural quotient map $\wh{X\ot_{r,p,q} Y}\to X\wh\ot_{r,p,q} Y$ is injective (recall that the last space is a subspace of the projective product $X\wh\ot Y).$ \smallskip {\it Remark 1.2.}\, For an "operator" situation, see Corollary 2.1 below % c1 to find and (for $1\le p,q,\le \infty)$ \cite[pp. 249-251]{PiOP}. %!!!ref \smallskip \begin{comment} ??? {\bf Proposition 2. --- NONCLEAR!!!}\, The natural map $Y\widehat\otimes_{r,p,q} X\to Y\widehat\otimes_{r,p,q} X^{**}$ is an isometric imbedding (and, in particular, one-to-one). %\noindent {\it Proof}.\, As was mentioned in Section 1, Remark 1, (iii), % previous!!! the dual space to $Y\widehat\otimes_{r,p,q} X$ is the space $\Pi_{\infty,p,q}(X,Y^*).$ ... ... ... \end{comment} \bigskip \centerline{\bf Approximation properties} \bigskip We begin with the main definition. %\smallskip %{\bf General information.} \small {\bf Definition 2.1.}\, A Banach space $X$ has the approximation property $AP_{r,p,q}$ if for every Banach space $Y$ the canonical mapping $Y\wh\ot_{r,p,q} X\to L(Y^*,X)$ is one to one. \medskip {\bf Proposition 2.1.}\, The following conditions are equivalent: 1)\, $X$ has the $AP_{r,p,q}.$ 2)\, For every Banach space $W$ the natural map $W^*\wh\ot_{r,p,q} X\to L(W,X)$ is one-to-one. 3)\, The natural map $X^*\wh\ot_{r,p,q} X\to L(X):=L(X,X)$ is one-to-one. \small {\it Proof.}\, Implications $1)\implies 2) \implies 3)$ are evident. $3)\implies 1).$\, %The proof repeats word for word the proof of Proposition 6.1 from \cite{Trend} %!!! %(instead of $\wh\ot_{\alpha}$ and $Y^*$ there we have to put now the tensor products %$\wh\ot_{r,p,q}$ and $Y$ respectively). %\centerline {\bf part 1 FOR ARXIVE} %For the sake of completeness, we give the proof here. Suppose that the canonical map $X^*\widehat\otimes_{r,p,q} X\to L(X)$ is one-to-one, but there exists a Banach space $Y$ such that the natural map $Y\widehat\otimes_{r,p,q} X\to L(Y^*, X)$ is not injective. Let $z\in Y\widehat\otimes_{r,p,q} X$ be such that $z\neq0$ and the associated operator $\widetilde z$ is a zero operator. Then we can find an operator $V$ from $L(Y,X^*)$ (the dual space to the projective tensor product $Y\widehat\otimes X)$ so that $\operatorname{trace}\, V\circ z^t=1,$ where, as usual, $z^t$ is the transposed tensor element, $z^t\in X\widehat\otimes Y.$ Since $V\circ z^t\in X\widehat\otimes X^*$ and $\operatorname{trace}\, V\circ z^t=1,$ the tensor element $(V\circ z^t)^t$ (which, evidently, belongs to $X^*\widehat\otimes_{r,p,q} X)$ is not zero. Contradiction. %\centerline {\bf end part 1 FOR ARXIVE} \medskip {\it Remark 2.1}.\ One can introduce also (in a similar way) some new notions of the approximation properties by using the Laprest\'e tensor products. We do not consider these properties here because we do not know how to work with the tensor products if their Banach duals do not separate points. \medskip The following assertion is an analogue of \cite[Prop. 6.2]{Trend}. %Again, %!!!Ref Its proof is contained in the corresponding proof of Proposition 6.2 from \cite{Trend}. %!!! But since a situation now is a little bit different from the one there (quasi-norms are here not "selfadjoint"), we present a proof here. %"self..."? %(essential different from the one in [rei Holl]). %!!! \medskip {\bf Proposition 2.2.}\, If $X^*$ has the $AP_{r,p,q},$ then $X$ has the $AP_{r,q,p}.$ \small %\noindent {\it Proof}.\, We will use Proposition 1. As it is known \cite{Gr}, the projective tensor product %!!!ref $X^*\widehat\otimes X$ is a Banach subspace of the tensor product $X^*\widehat\otimes X^{**}.$ The tensor product $X^*\widehat\otimes_{r,q,p} X $ is a linear subspace of $X^*\widehat\otimes X,$ as well as $X^*\widehat\otimes_{r,q,p} X^{**}$ is a linear subspace of $X^*\widehat\otimes X^{**}.$ Therefore, the natural map $X^*\widehat\otimes_{r,q,p} X \to X^*\widehat\otimes_{r,q,p} X^{**} $ is one-to-one. Now if $X^*$ has the $AP_{r.p.q},$ then the canonical map $X^{**}\widehat\otimes_{r,p,q} X^* \to L(X^*,X^*)$ is one-to-one. Since we can identify the tensor product $X^{**}\widehat\otimes_{r,p,q} X^*$ with the tensor product $X^{*}\widehat\otimes_{r,q,p} X^{**},$ it follows that the natural map $X^*\widehat\otimes_{r,q,p} X \to L(X,X)$ is one-to-one. Thus, if $X^*$ has the $AP_{r,p,q},$ then $X$ has the $AP_{r,q,p}.$ \medskip % We will use Lemma 4 in the proof of Theorem 1 in the next section. % \smallskip %%%%%%%% Zamechanie o tochnosti - perepravit' !!! % \noindent {\it Remark 2.2.}\, The inverse statement is not true. Some examples are given in \cite[Remark 6.1]{Trend}. %!!!Ref % See also ... below. %!!! below - corrol-s for some r,p,q: [r,p]... \medskip {\it Remark 2.3.}\, (i)\, From the proof it follows that: {\it For any $X$ and $Y$ the natural map $X\widehat\otimes_{r,q,p} Y \to X\widehat\otimes_{r,q,p} Y^{**}$ is one-to-one.} (ii)\, On the other hand: {\it For any $X$ and $Y$ the natural map $\widehat{X\otimes_{r,q,p} Y} \to \widehat{X\otimes_{r,q,p} Y^{**}}$} is an isometric embedding. To prove this, it is enough to apply Principle of Local Reflexivity (see e.g. \cite[E.3.1]{PiOP}) as it was done in \cite[18.1.12]{PiOP} for the case $1\le p,q\le\infty.$ \medskip Recall that a linear map $T: X\to Y$ is called $(r,p,q)$-nuclear if it has a representation $T= \sum_{k=1}^\infty \al_k\, \ y_k,$ where $(\al_k)\in l_r,$ $(x'_k)\in l_{w,p}(X^*)$ and $(y_k)\in l_{w,q}(Y).$ Every such a map is continuous. The space $N_{r,p,q}(X,Y)$ of all $(r,p,q)$-nuclear operators from $X$ to $Y$ can be considered as a quotient of the tensor product $X^*\wh\ot_{r,p,q} Y$ (as well as a quotient of $\wh{X^*\ot_{r,p,q} Y})$ by the kernel of the natural map $X^*\wh\ot_{r,p,q} Y \to L(X,Y).$ We equip this space with the induced quasi-norm $(\beta$-norm) denoted by $\nu_{r,p,q}.$ If the corresponding quotient map has a trivial kernel, then we write $N_{r,p,q}(X,Y)= X^*\wh\ot_{r,p,q} Y$ (respectively, $N_{r,p,q}(X,Y)= \wh{X^*\ot_{r,p,q} Y}).$ Thus, $X$ has the $AP_{r,p,q}$ iff for every Banach space $Y$ the equality $N_{r,p,q}(Y,X)= Y^*\wh\ot_{r,p,q} X$ holds. \small {\it Remark 2.4.}\, 1)\, If $t\in (0, +\infty],$ then $N_{t,p,q}(X, Y)$ (the space of $(t,p,q)$-nuclear operators) can be defined by the analogues way: an operator $T: X\to Y$ is $(t,p,q)$-nuclear, if it can be written in the form $T = \sum_{k=1}^\infty \al_k\, \ y_k,$ where $(\al_k)\in l_t,$ $(x'_k)\in l_{w,p}(X^*)$ and $(y_k)\in l_{w,q}.$ The quasi-norm $||\cdot||_{N_{t,p,q}}$ is defined for $T$ as $\inf ||(\al_k)||_t ||(x'_k)||_{w,p} ||(y_k)||_{w,q},$ where infimum is taken over all appropriate representations of $T.$ If $1/t + 1/p + 1/q=1,$ then it is a norm. %We will use these spaces in Section ... %!!! where - write 2)\, In notation, we follow J.-T. Laprest\'e \cite{166}, only changing a triple %!!! ref $(p,r,s)$ there to $(r,p,q)$ here (see also \cite{PiOP} ; nota bene: A. Pietsch \cite[18.1]{PiOP} uses different notations %!!! ref for this class of operators and considers the cases where $1\le p,q\le\infty.)$ \medskip It follows from Proposition 1.1: \medskip {\bf Proposition 2.3}\, %In conditions of Proposition 0, Let 1)\, $0 Y as above and below DO NOT WANT! %1)\, The tensor product $\wh{\cdot\ot_{r,p,2}\cdot}$ is injective, i.e. if $i: Y\to W$ is an isometric imbedding and $z\in \wh{X\ot_{r,p,2}Y},$ then $\mu_{r,p,2}(z)=\mu_{r,p,2}(i\circ z).$ %2)\, %The tensor product $\wh{\cdot\ot_{r,2,q}\cdot}$ is surjective, i.e. %!!! может, есть что подобное? %if $Q: V\to X$ is a quotient map and $z\in \wh{X\ot_{r,2,q}Y},$ %then $\mu_{r,2,q}(z)=\mu_{r,2,q}(i\circ z).$ \small {\it Proof}.\, %1)\, It is clear that $\mu_{r,p,2}(z)\ge\mu_{r,p,2}(i\circ z).$ Let $\e>0$ and $\sum_{k=1}^N \al_k x_k\ot \ffi_k$ be a finite representation of $i\circ z$ %in $\wh{X\ot_{r,p,2}W}$ such that in $X\ot W$ such that $||(\al_k)||_r ||(x_k)||_{w,p} ||(\ffi_k)||_{w,q}\le (1+\e) \mu_{r,p,2}(i\circ z).$ Define an operator $S\in L(l^N_2,W)$ and a tensor element $z_0\in \wh{X\ot_{r,p,2}l^N_2}$ by $S = \sum e'_k\ot \ffi_k$ and $z_0=\sum \al_k x_k\ot e_k.$ Let $E:=\ove{\wt{z_0}(X^*)}\subset l_2^N$ and $P: l_2^N\to l_2^N$ be an orthogonal projector from $l_2^n$ onto $E.$ Then $S\circ z_0=i\circ z,$ $\wt{SP\circ z_0}=\wt{i\circ z}$ and $SP(l_2^N)\subset i(Y).$ It follows that $z= (i^{-1}|_{i(Y)}SP)\circ z_0= \sum \al_k x_k\ot i^{-1}|_{i(Y)}SPe_k$ (as the elements of $X\ot Y)$ and $$\mu_{r,p,2}(z)\le ||S||\, ||(\al_k)||_r ||(x_k)||_{w,p}= ||(\al_k)||_r ||(x_k)||_{w,p} ||(\ffi_k)||_{w,q}\le (1+\e) \mu_{r,p,2}(i\circ z).$$ Therefore, the natural map $\id_X\ot i: X\ot Y\to X\ot W$ is an isometric imbedding and it extends to the isometry $\id_X\ove{\ot} i: \wh{X\ot_{r,p,2} Y}\to \wh{X\ot_{r,p,2} W}.$ \medskip % 2)\, {\it Remark 2.5.% to3 }.\, %!!! to3 and below It follows from Lemma 2.1 %to3 and from definitions of $\wh\ot_{r,p,2}$ (and $N_{r,p,2})$ that $\wh\ot_{r,p,2}$ (and $N_{r,p,2})$ are injective (consider the quotient maps $\wh{\ot_{r,p,2}}\to \wh\ot_{r,p,2}\to N_{r,p,2}$). On the other hand, the injectivity of $\wh\ot_{r,p,2}$ (and $N_{r,p,2})$ can be proved in the same way as above by consideration the infinite representations of $z, \wt z$ (instead of finite ones) in the given proof. Also, we see from the proof that if $\sum_{k=1}^\infty \al_k x_k\ot \ffi_k$ is a representation of $i\circ z$ in $\wh{X\ot_{r,p,2}W}$ (see \cite[Proposition 1.3, p. 52]{166}, then the corresponding representation for $z$ in $\wh{X\ot_{r,p,2} Y}$ can be taken of the type $\sum_{k=1}^\infty \al_k x_k\ot y_k$ with $||(y_k)||_{w,2}\le ||(\ffi_k)||_{w,2}.$ The same is true for $\wh\ot_{r,p,2}$ and $N_{r,p,2}.$ \medskip % repr. Lapr-Th on ..-nucl. [166, Proposition 1.3, p. 52]. The first part of the following fact is partially known (cf. \cite[18.1.15-18.1.16]{PiOP} for $1\le p,q\le\infty).$ \medskip {\bf Proposition 2.4.}\, For any Banach spaces $X,Y$ the equalities $$ N_{r,p,2}(Y,X)= Y^*\wh\ot_{r,p,2} X = \wh{Y^*\ot_{r,p,2} X} \ \mathrm{ and } \ N_{r,2,q}(Y,X)= Y^*\wh\ot_{r,2,q} X = \wh{Y^*\ot_{r,2,q} X} $$ hold (with the same quasi-norms). In particular, every Banach space has the $AP_{r,p,2}$ and the $AP_{r,2,p}.$ \small % This fact is essentially known (see [PiOP, 18.1.15-18.1.16] for $1\le p,q\le\infty). %\centerline {\bf part 2 FOR ARXIVE} \medskip {\it Proof}.\, As is known, the operator ideal $N_{r,p,2}$ is injective, (see \cite[18.1.8]{PiOP} %%!!!Ref for the case $1\le p,q\le\infty);$ apply (factorization) Theorem 2.5 \cite{166} %!!!ref in other cases). %!!!ref I.e., if $X\subset G,$ $T\in L(Y,X)$ and $T\in N_{r,p,2}(Y,G),$ then $T\in N_{r,p,2}(Y,X)$ (with the same quasi-norm). Also, the tensor product $\wh{\cdot\ot_{r,p,2} \cdot}$ is injective too (Lemma 2.1)%to3) %apply the same factorization theorem to "finite" tensors $z\in E\ot F).$ %!!! to3 Now, let $z\in \wh{Y^*\ot_{r,p,2} X}$ and $i: X\to L_\infty$ be an isometric embedding of $X$ into an $L_\infty$-space. Since $L_\infty$ has the MAP, $\nu_{r,p,2}(\wt{i\circ z})= \mu_{r,p,2}(i\circ z)$ (see Corollary 2.1). %!!! find: c1 - CHECK IT Hence, $\mu_{r,p,2}(z)=\mu_{r,p,2}(i\circ z)=\nu_{r,p,2}(\wt{i\circ z})= \nu_{r,p,2}(i \wt z)\le \nu_{r,p,2}(\wt z)\le \mu_{r,p,2}(z).$ To get the last two equalities it is enough to apply the surjectivity of %!!! details! or else... the operator ideal $N_{r,2,q}$ and Corollary 2.1 (second part), by using the same idea as above, or just to apply Lemma 2.1 %to3 and Remark 2.5 %to3 (second part): %!!! Rem to3 - лишнее? Take $z\in \wh{Y^*\ot_{r,2,q} X}$ and a quotient map $Q: L_1\to Y.$ Considering $(z\circ Q)^t$ as an element of $\wh{X\ot_{r,q,2} L_\infty},$ we get: $$ \nu_{r,q,2}(\wt{Q^*\circ z^t})= \mu_{r,q,2}(Q^*\circ z^t) =\mu_{r,q,2}(z^t)= \mu_{r,2,q}(z). $$ But $\nu_{r,q,2}(\wt{Q^*\circ z^t})\le \nu_{r,2,q}(\wt{z\circ Q})\le \nu_{r,2,q}(z);$ thus, $ \mu_{r,2,q}(z)\le \nu_{r,2,q}(z).$ \medskip %{\bf ...} , %\centerline {\bf end part 2 FOR ARXIVE} % \small % [Pi consid. only (1,2,\infty) 27.4.10 ---later on Gr-Lids !!! {\it Remark 2.6.}\, The fact that every $X$ has the $AP_{1,2,\infty}$ is essentially contained in \cite[27.4.10, Proposition]{PiOP}. It is strange, but it seems that a corresponding fact for $AP_{1,\infty,2}$ appears here for the first time. Note that this fact follows also from the preceding by virtue of Proposition 2.2: if every $X$ has the $AP_{1,2,\infty},$ then $X^*$ possesses this property, and by Proposition 2.2 $X$ has the $AP_{1,\infty,2}.$ \medskip Many of the above approximation properties were considered earlier, e.g. in the papers \cite{RQ, Rs, Trend} etc: %!!!refs - below (i)\, For $p=q=\infty,$ we get the $AP_r$ from \cite{Rs, Trend}. %!!! (ii)\, For $p=\infty,$ we get the $AP_{[r,q]}$ from \cite{RQ, Trend}. %!!! (iii)\, For $q=\infty,$ we get the $AP^{[r,p]}$ from \cite{RQ, Trend}. %!!! %[RQ] O. I. Reinov, Q. Latif, Distribution of eigenvalues of nuclear operators and %Grothendieck-Lidski type formulas, Journal of Mathematical Sciences, Springer, %Vol. 193, No. 2, August, 2013, 312-329. % %[Rs] O.I. Reinov, On linear operators with s-nuclear adjoints, $0 < s \le 1,$ J. Math. %Anal. Appl. 415 (2014) 816-824. % %[Rrp] Oleg Reinov, Some Remarks on Approximation Properties, with Applications, %Ordered Structures and Applications: Positivity VII Trends in Mathematics, %371-394. 2016 Springer International Publishing \smallskip Following notations from \cite{Trend} (see also \cite{RQ}), we denote %!!!ref $N_{r,\infty,\infty}$ by $N_r,$ $N_{r,\infty,q}$ by $N_{[r,q]},$ $N_{r,p,\infty}$ by $N^{[r,p]},$ $\wh\ot_{r,\infty,\infty}$ by $\wh\ot_{r},$ $\wh\ot_{r,\infty,q}$ by $\wh\ot_{[r,q]},$ $\wh\ot_{r,p,\infty,}$ by $\wh\ot^{[r,p]}.$ The corresponding notations are used also for the $AP_{r,p,q}$ (see above (i)--(iii)). Almost all the information about Banach spaces without (or with) the properties $AP_r,$ $AP_{[r,q]}$ and $AP^{[r,p]}$ which is known to us by now, can be found in \cite{RQ, Rs, Trend}. Other results in this direction are the subject of the forthcoming paper of the author. \bigskip \begin{comment} {\bf In $L_p$-spaces.}%Approximation properties in spaces of type and cotype.} \medskip \bigskip %% SECTION BIG NEXT \centerline{SECTION II.} \bigskip $$- - - $$ {\bf On $T^*$ and $T$ for (r,p,q)-nuclerity.} It is clear that we have: \small {\bf Proposotion D.}\, If $T\in N_{r,p,q},$ then $T^*\in N_{r,q,p}.$ \small --- [(r,p,2) ]... %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{comment} %%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{00} \bibitem{Gr} A.~Grothendieck: \textit{Produits tensoriels topologiques et \'espaces nucl\'eaires}, {Mem. Amer. Math. Soc.} \textbf{16}(1955). \bibitem{RQ} O. I. Reinov, Q. Latif, \textit{Distribution of eigenvalues of nuclear operators and Grothendieck-Lidski type formulas}, Journal of Mathematical Sciences, Springer, Vol. 193, No. 2, August, 2013, 312-329. \bibitem{Rs} O.I. Reinov, \textit{On linear operators with s-nuclear adjoints, $0 < s \le 1,$} J. Math. Anal. Appl. 415 (2014) 816-824. \bibitem{Trend} Oleg Reinov, \textit{Some Remarks on Approximation Properties with Applications}, Ordered Structures and Applications: Positivity VII Trends in Mathematics, 371-394, 2016 \bibitem{166} Lapreste, J. T.: \textit{Op\'erateurs sommants et factorisations \`a travers les espaces $L_p$}, Studia Math. 57(1976)47-83 \bibitem{PiOP} A. Pietsch: \textit{Operator Ideals}, North Holland (1980). \bibitem{Pis} G. Pisier, \textit{The volume of convex bodies and Banach space geometry}, Cambridge tracts in mathematics {\textbf 94}, Cambridge University Press 1989. \end{thebibliography} \end{document} %%%%%%%%%%% {\bf Lemma 2.7.}\, % {Lemma 2.}\, %!!! rename it! (i)\, Let $z\in \wh{Y^*\ot_{r,p,2} X}.$ If $i: X\to G$ be an isomorphic embedding such that $\nu_{r,p,2}(\wt{i\circ z})>0,$ then $\nu_{r,,p,q}(\wt z)\ge0.$ ii)\, Let $z\in \wh{Y^*\ot_{r,2,q} X}.$ If $Q: W\to Y$ be a surjection such that $\nu_{r,2,q}(\wt{i\circ z})>0,$ then $\nu_{r,,2,q}(\wt z)\ge0.$ {\it Proof}.\, (i)\, Fix $\e>0$ and consider the operator $i\circ \wt z.$ By Theorem 2.5 [Lapr], %!!! $i\circ\wt z=BA,$ where $B\in L(l_2,G), ||B||=1,$ $A\in N_{r,p,2}(Y,l_2)$ and $0<\nu_{r,p,2}(A)\le \nu_{r,p,2}(\wt{i\circ z})+\e.$ Let $P: l_2\to H:=B^{-1}(\over{BA(Y)}$ be an orthogonal projector. We have $\wt{i\circ z}= B|_{H}PA,$ $B|_{H}PA(Y)\subset i(X)$ and $\wt z= i^{-1}|_{i(X)} B|_{H}PA.$ Hence, 0<\nu_{r,p,2}(\wt {i\circ z}) %%%% ... We apply Lemma 2.7: ... The proof is essentially known (cf. \cite[18.1.15 and 18.1.16]{PiOP} for $1\le p,q\le\infty).$ Let $T\in N_{r,p,2}(Y,X)$ and $i: X\to L_\infty(\mu)$ be an isometric embedding. By Theorem 2.5 [Lapr], %!!! $T=BA,$ where $B\in L(l_2,X)$ and $A\in N_{r,p,2}(Y,l_2).$ {\bf Remark.}\, ... ... In the cases where $p=2,$ or $q=2$ Pietsch.... %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{00} \bibitem{1} J. Bergh, J. L\"ofstr\"om: \textit{ Interpolation spaces}, Berlin, Heidelberg, New York: Springer-Verlag (1976). \bibitem{2} J. Bourgain, O.I. Reinov: \textit{On the approximation properties for the space $H^\infty,$} Math. Nachr. \textbf{122}, No. 2 (1985), 19-27. \bibitem{3} A.M. Davie: \textit{The approximation problem for Banach spaces}, Bull. London Math. Soc. \textbf{5} (1973), 261--266. \bibitem{4} J. Diestel, H. Jarchow, A. Tonge: \textit{Absolutely summing operators}, Cambridge Univ. Press, 1995. \bibitem{5}N. Dunford, J. Schwartz: \textit{Linear operators, Part I}, Interscience, New York (1958). \bibitem{6} P. Enflo: \textit{A counterexample to the approximation property in Banach spaces}, Acta Math. \textbf{130} (1973), 309-317. \bibitem{7} A.~Grothendieck: \textit{Produits tensoriels topologiques et \'espaces nucl\'eaires}, {Mem. Amer. Math. Soc.} \textbf{16}(1955). \bibitem{8} A. Hinrichs, A. Pietsch: $p$-nuclear operators in the sense of Grothendieck, Math. Nachr. \textbf{283}, No. 2 (2010), 232--261. \bibitem{9} M. J. Kadec, M. G. Snobar: \textit{Certain functionals on the Minkowski compactum} (Russian), {Mat. Zametki} \textbf{10} (1971), 453-458. \bibitem{10} H. Konig: \textit{Eigenvalue distribution of compact operators}, Birkh\"auser (1986). \bibitem{11} H. Konig: \textit{On the eigenvalue spectrum of certain operator ideals}, Coll. Math. \textbf{44} (1981), 15--28. \bibitem{12} V.\,B.~Lidski\v{\i}: \textit{Nonselfadjoint operators having a trace} (Russian), {Dokl. Akad. Nauk SSSR} \textbf{125} (1959), 485--487. \bibitem{13} J. Lindenstrauss: \textit{On James' paper "Separable Conjugate Spaces"}, Israel J. Math. \textbf{9} (1971), 279--284. \bibitem{14} J. Lindenstrauss, L. Tzafriri: \textit{Classical Banach spaces, vol.1: Sequence spaces}, Berlin-Heidelberg-New York (1977). \bibitem{15} A. Pietsch: \textit{Eigenwertverteilungen von Operatoren in Banachr\"aumen}, Hausdorff-Festband: Theory of sets and topology, Berlin: Akademie-Verlag (1972), 391-402. \bibitem{16} A. Pietsch: \textit{Operator Ideals}, North Holland (1980). \bibitem{17} A. Pietsch: \textit{Eigenvalues and s-numbers}, Cambridge Univ. Press (1987). \bibitem{18} A. Pietsch: \textit{History of Banach spaces and linear operators}, Birkh\"auser (2007). \bibitem{19} G. Pisier: \textit{Estimations des distances {\`a} un espace euclidien et des constantes de proj{\'e}ction des espaces de Banach de dimensions finie}, Seminaire d'Analyse Fonctionelle 1978-1979, Centre de Math., Ecole Polytech., Paris (1979), expos{\'e} 10, 1--21. \bibitem{20} O.I. Reinov: \textit{A simple proof of two theorems of A. Grothendieck} (Russian), Vestn. Leningr. Univ. \textbf{7} (1983), 115-116. \bibitem{21} O.I. Reinov: \textit{Disappearance of tensor elements in the scale of p-nuclear operators} (Russian), Theory of Operators and Theory of Functions, Leningrad , LGU, vol. 1 (1983), 145-165. \bibitem{22} O.I. Reinov: \textit{A survey of some results in connection with Grothendieck approximation property}, Math. Nachr. \textbf{119} (1984), 257-264. \bibitem{23} O.I. Reinov: \textit{Approximation properties $AP_s$ and p-nuclear operators (the case $0 0$ one can find a finite rank operator $R$ in $X$ such that $\sup_{x\in K}\, ||Rx-x||\le \varepsilon.$ The property is so important that we can find its applications in a great number of papers %in different directions of (e. g.) devoted to the theory of operators in Banach spaces. Let us reformulate (following [7]) %!!!R the definition of the $AP$ in terms of tensor products: $X$ has the $AP$ iff the natural mapping $j:\, X^*\widehat\otimes X\to L(X,X)$ is one-to-one (here by "natural map" we mean the unique extension of the natural linear inclusion $X^*\otimes X\to L(X,X)$ from the normed space $(X^*\otimes X, ||\cdot||_{\land})$\, (i.e. with the projective norm) to the completion $X^*\widehat\otimes X).$ This fact becomes evident %is immediate if we note that the map $j$ is one-to-one iff the closure of $X^*\otimes X$ in $L_c(X,X)$ is the whole space $L(X,X)$ (or, what is the same, the identity map $\operatorname{id}_X$ is in this closure). The image of the tensor product $X^*\widehat\otimes X$ in $L(X,X)$ is, by definition, the space $N(X,X)$ of nuclear operators in $X$ (with the norm induced from $X^*\widehat\otimes X).$ Thus, $X$ has the $AP$ iff (as we can write) $N(X,X)= X^*\widehat\otimes X.$ On the space $X^*\otimes X,$ the usual linear functional "trace" is defined which is continuous on the normed space $(X^*\otimes X, ||\cdot||_{\land}).$ After extension to the projective tensor product $X^*\widehat\otimes X,$ this linear functional is still bounded and it can be seen that a Banach space $Y$ has the $AP$ iff every tensor element $z\in Y^*\widehat\otimes Y$ which generates a 0-operator in $L(Y,Y)$ has the property that $\operatorname{trace}\, U\circ z=0$ for every $U\in L(Y,Y^{**}).$ %is zero itself. In Chapter II of [7], %!!!R A. Grothendieck has generalized the notion of nuclear operators and also considered the more general tensor products: In the terminology of [7], %!!!R an element of $X^*\widehat\otimes Y$ is said to be a "noyau de Fredholm de puissance p.\`eme sommable" $(p\in (0,1]),$ if it is of the form $\sum_{i} \lambda_i\, x'_i\otimes y_i,$ where $(\lambda_i)\in l_p$ and $(x_i)$ (resp. $y_i)$ is a bounded sequence in $X^*$ (resp. $Y).$ We will use the notation $X^*\widehat\otimes_p Y$ for the corresponding tensor product (in [7], it is denoted by $X^* \overset{(p)}{\otimes} Y).$ This is a linear subspace of the projective %!!!R tensor product $X^*\widehat\otimes Y,$ and with the natural metric (see [7], Chapter II, \S1) it is %!!!R a complete metric space. If $z\in X^*\widehat\otimes_p Y,$ then the associated operator $\widetilde z$ from $X$ to $Y$ is called (by A. Grothendieck) as "une application de puissance p.\`eme sommable". The natural inclusion $X^*\widehat\otimes_p Y \hookrightarrow X^*\widehat\otimes Y$ is one-to-one for any pair of Banach spaces $X$ and $Y$ (a priori, it is not evident; see [7], Chapter II, \S1 for an explanation). %!!!R One of the interesting questions considered in [7], is the connection between %!!!R the "order" of a tensor element $z\in X^*\widehat\otimes_p Y$ and the "order" of the sequence of all eigenvalues of the corresponding operator $\widetilde z$ (evidently, $\widetilde z$ is compact). Among the results in this direction, let us mention only the following facts (which were obtained in [7], Chapter II, \S1, Section 4): %!!!R Let $X$ and $Y$ be Banach spaces and $00)$ in $X,$ if there exists a linear continuous projection $P$ from $X$ onto $E$ such that $||P||\le b.$ As usual, if $p\in [1,\infty],$ then $p'$ is the conjugate exponent: $1/p+1/p'=1.$ %\begin{proposition}{\it Let $X$ be a Banach space and $10$ such that every finite dimensional subspace $E$ of $X$ is $D_{p.q}\, ( {\operatorname{dim} } E)^{1/p-1/q}$-complemented in $X.$ %} %\end{proposition} Recall also the well known general fact (due to M. J. Kadec and M. G. Snobar [9]; see also %!!!Ref i nizhe [16, 28.2.6. Lemma]): in any Banach space every $n$-dimensional subspace is $n^{1/2}$-complemented. %% Our main refer.... %как в тексте абзац %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BEGIM.TEX %\vskip 0.3cm Our main reference is [16]. All the notions, notations and facts, %!!!Ref i 4-ja nizhe given here without any explanation, %we use without any reference, can be found in [1, 5, 8, 14, 16, 17, 18]. %%%%%%%%%%%%%%%%%%%%%%%%% Content \medskip {\bf Contents.}\, In Section I we reformulate the definition of Grothendieck approximation property in terms of 0-sequences that leads us to the consideration of more general approximation properties $\widetilde{AP_s}$ for $s\in(0,1].$ We define them in terms of approximation of the identity maps in Banach spaces by finite rank operators on $l_p$-sequences $(p$ depends on $s).$ It seems that such properties, for the first time, were considered by the author in [22] (cf. Lemma 2.1 there). The simplest examples %!!!Ref of Banach spaces possessing the properties of such a kind are subspaces of quotients of $L_p$-spaces. In Section II we reformulate the $\widetilde{AP_s}$ in terms of tensor products showing that that $\widetilde{AP_s}=AP_s,$ where $AP_s$ is the approximation property of order $s$ introduced in the author paper [21] and investigated also in [2], [23], [27]. %!!!Ref In Section III we introduce and investigate new notions of the approximation properties $AP_{t;p,r}$ and $AP_{(r,w)}$ defined by some "Lorentz tensor products" (tensor products generated by Lorentz sequence spaces). These notions are new and considered here for the first time. In particular, we obtain a characterization of the $AP_{(r,1)}$ in terms of approximation of the identity maps by finite rank operators on some (Lorentz) 0-sequences (Theorem 3.3). In Section IV, with the help of a theorem of M.C. White [32], we prove Theorem 4.1 %!!!Ref i nizhe 1-2 strok which gives us sufficient conditions for the famous Grothendieck-Lidski{\v{\i}} trace formula to be valid for certain quasi-Banach operator ideals. We present an application of the previous results about approximation properties to some eigenvalue problems. Theorem 4.1 is applied then to the case of $(2/3,1)$-nuclear operators (related to the Lorentz space $l_{2/3\,1}).$ %!!!?? related? In Section V the results of Sections I about subspaces of quotients of $L_p$-spaces together with White's theorem are applied for proving some more theorems concerning the distribution of eigenvalues of the nuclear operators. We give new relatively simple proofs of some recent results from the papers [25] and [28]. %!!!Ref 2nd - to appear!!! In Section VI we prove two statements about the approximation properties considered in Sections I--V. For instance, it is well known that if $X^*$ has the $AP$ of Grothendieck, then $X$ has the $AP$ too. We show that the same is true for all the natural approximation properties considered here, such that $AP_s, AP_{(r,w)}$ etc. In Section VII we introduce, following [24] and [26], two more notions of the %!!!Ref i nizhe v abzatce approximation properties by using the spaces of so-called $(r,p)$-nuclear operators (a partial case of a class of $(s,q,t)$-nuclear operators from [16, 18.1]). Two theorems about eigenvalues of the $(r,p)$-nuclear operators are proved. In these theorems, trace formulas of Grothendieck-Lidski{\v{\i}}type are established for the cases where $1/r-1/p=1/2.$ The first one was proved in [24] and [26] with the help of Fredholm Theory; the second theorem (Theorem 7.3) was obtained before by the same authors, again by using Fredholm Theory (but its proof was unpublished). Here, the different (more simple) method is used. Firstly, we show that every Banach space has the corresponding approximation properties $AP_{[r,p]}$ and $AP^{[r,p]}.$ After this, we obtain the eigenvalue results by using simple arguments. Finally, in Section VIII, examples are given (they are taken from the paper [26]). %!!!Ref The examples give a possibility to conclude that all the positive results of Sections I--VII concerning the approximation properties and trace formulas are sharp. %%%%%%%%%%%%%%%%%%%%%% sections 1-... %%%%%%%%%%%%%%%%%%% I-...-VIII %777 - begin to change refrnc No's \medskip {\bf I.}\ It is well known that every compact subset of a Banach space is contained in the closed % vstavil i dal ssylki 11 Октябрь 2014 г. 22:48:39 convex hull of a sequence converging to 0 (see, e.g., [7], p. 112 in Ch.I, Lemme 12, or [14], p. 30, Proposition 1.e.2). Therefore, the Grothendieck approximation property for a Banach space $X$ can be defined as follows: $X$ has the $AP$ iff for every sequence $(x_n)_{n=1}^\infty\subset X$ tending to zero, for any $\varepsilon>0$ there exists a finite rank (continuous) operator $R$ in $X$ such that for each $n\in\Bbb N$ one has $||Rx_n-x_n||\le\varepsilon.$ Consider a natural question: for which sequences $(x_n)\in c_0(X),$ under some additional assumptions, the identity map $\operatorname{id}_X$ can be approximated by finite rank operators, as above, and which of those conditions are sharp (or, if one wishes, optimal)? One of the simplest fact (we think, known for more than 30 years) is that $(*)$\, if $(x_n)\in l_2(X),$ \, $X$ is any, then the answer is positive. Here is a reason of this: Assuming $||x_n||\searrow 0,$ take any $N\in \Bbb N$ and consider the linear span $E_N:=\operatorname{span}[x_n]_1^N$ as a subspace of $X.$ Define, fixing an $\varepsilon>0,$ a finite rank operator $R$ to be a projection from $X$ onto $E_N$ whose norm $\le \sqrt N.$ Now if $N$ is such that, for every $n\ge N,$ we have $||x_n||\le \frac {\varepsilon}{\sqrt N+1},$ then $$ ||Rx_n-x_n||=0 \ \text{ if } \ n\le N, $$ and $$ ||Rx_n-x_n||\le (||R||+1)\, ||x_n|| \le \varepsilon \ \text{ if } \ n\ge N. $$ Of course, instead of $(*)$ we can consider the statement $(**)$\, if $(x_n)\in l_{2,\infty}^0(X) $ \, (Lorentz space with "o" small --- %!!! popravil 11 Октябрь 2014 г. 22:41:11 $l^{min}_{(2, \infty)}(X)$ in notations of [16, 13.9.3 Remark]), $X$ is any, then the answer is affirmative. The idea of the above proof is very simple and can be applied in some more general situations. For instance, every subspace of finite dimension $n$ of an $L_p$-space is $n^{|1/2-1/p|}$-complemented in that $L_p$-space. So, if $p\in [1,\infty],$ $\alpha=|\frac{1}{2}-\frac1p|$ and $X$ is a subspace of an $L_p$-space, then $(***)$\, for every sequence $(x_n)\in l_{q,\infty}^0(X),$ where $1/q=\alpha,$ the answer is affirmative. \smallskip {\it Remark \rm1.1}:\, About sharpness: it will be discussed below. \smallskip {\it Remark \rm1.2}:\, The statement $(***)$ has, as a matter of fact, the following quantitative aspect: Given $\alpha\in [0,1/2]$ and a Banach space $X$ with the property that every finite dimensional subspace $F$ of $X$ is contained in a finite dimensional subspace $E\subset X,$ which in turn is $C\, (\operatorname{dim} F)^\alpha$-complemented in $X,$ we have $(***)'$\, for every sequence $(x_n)\in l_{q,\infty}^0(X),$ where $1/q=\alpha,$ for any $\varepsilon>0$ there is a finite rank operator $R$ in $X$ so that $\sup_n ||Rx_n-x_n||\le\varepsilon.$ Particular cases: \noindent (i)\, $q=2$ and $\alpha=1/2$ or $q=\infty$ and $\alpha=0;$ %(i. e.,\, "$X$ is any Banach space" or "$X$ is isomorphic to a Hilbert space"); \noindent (ii)\, $(x_n)\in l_q(X), q\in [2,\infty). $ %, or $(x_n)\in c_0(X), q=\infty$ [Hilbert case]. \smallskip For a while let us introduce the notions of the corresponding approximation properties for a Banach space $X$ (taking into account that the possibility of approximations on $c_0$-sequences by finite rank operators gives us the Grothendieck's approximation property $AP):$ Let $00$ there exists a finite rank operator $R\in X^*\otimes X$ such that $\sup_n ||Rx_n-x_n||\le \varepsilon.$ Trivially, e.g., $\widetilde{AP}_{s_2} \implies \widetilde{AP}_{s_1}$ if $s_1\le s_2.$ Thus, $\widetilde{AP}_1 (= AP)$ implies any $\widetilde{AP}_s.$ The statement $(*)$ (and $(**)$) says that every Banach space has the above property $\widetilde{AP}_{2/3}$ (and even the $\widetilde{AP}_{2/3,\infty}).$ The statement $(***)$ gives the corresponding result for $L_p$-subspaces. Moreover, the assertion mentioned in Remark 1.2, shows that, for instance, any subspace of any quotient (= any quotient of any subspace) of a Banach space of type 2 (resp., of cotype 2) and of cotype $p,$ \, $p\in[2,\infty)$ (resp., of type $p'),$ possesses the $\widetilde{AP}_s$ (even the $\widetilde{AP}_{s,\infty})$ with $1/s=1+|1/2-1/p|.$ \vskip 0.23cm {\bf II.}\ Let us recall that the notion of the $AP$ of Grothendieck can be reformulated in terms of the projective tensor products "$\widehat\otimes$". Namely, a Banach space $X$ has the $AP$ iff for every Banach space $Y$ the canonical (natural) mapping $Y^*\widehat\otimes X \to L(Y,X)$ is one-to-one (or, what is the same, the natural mapping $X^*\widehat\otimes X\to L(X):=L(X,X)$ is injective). In [7], A. Grothendieck has considered also some other tensor products (linear subspaces %!!!R of "$\widehat\otimes$"'s), which we will denote by "$\widehat\otimes_s$" for $00.$ Then $$ 0<\operatorname{trace}\, R\circ z_0 = \sum_{k=1}^\infty \langle x'_k, x\rangle \langle x', x_k\rangle = \sum_{k=1}^\infty \lambda_k\,\langle U^*y'_k, x\rangle \langle x', x_k\rangle $$ $$= \langle\sum_{k=1}^\infty \lambda_k\, \langle Ux, y'_k\rangle x_k, x'\rangle = \langle x', \sum_{k=1}^\infty \lambda_k\, U^*y'_k(x) x_k\rangle =0. $$ \smallskip {\it Proof}\ of Theorem 2.1. We will use the assertion $(AP_s).$ $1) \implies 2).$\, Let $z\in X^*\widehat\otimes_s X$ and $\operatorname{trace}\, z=1.$ Write $z= \sum \lambda_k\, x'_k\otimes x_k,$ where the sequences $(x'_k)$ and $(x_k)$ are bounded and $(\lambda_k)\in l_s,$ $\lambda_k\ge0,$ $(\lambda_k)$ is non-increasing. Then $$ z=\sum_{k=1}^\infty (\lambda_k^s\, x'_k)\otimes (\lambda_k^{1-s}\, x_k) $$ (recall that $1/s=1+1/q;$ so $1-s=s/q).$ The sequence $(\lambda_k^{1-s}x_k)$ is in $l_q(X).$ By 1), for every $\varepsilon>0$ there exists a finite rank operator $R\in X^*\otimes X$ such that $||R(\lambda_k^{1-s}x_k)-\lambda_k^{1-s}x_k||\le\varepsilon$ for each $k\in \Bbb N.$ It follows that, for this operator $R,$ $$ |\operatorname{trace}\, (z- R\circ z)| = |\sum_{k=1}^\infty \langle\lambda_k^s x'_k, \lambda_k^{1-s} x_k- R(\lambda_k^{1-s} x_k)\rangle | \le \sum_{k=1}^\infty \lambda_k^s ||x'_k||\cdot \varepsilon\le const\cdot \varepsilon. $$ Hence, for small $\varepsilon>0$ we have that, for the operator $R\in X^*\otimes X,$ $$ |\operatorname{trace}\, R\circ z| \ge 1/2 $$ and therefore $z$ generates a non-zero operator $\widetilde z.$ \smallskip Before consider a proof of the implication $2)\implies 1)$ we will make some additional remarks. We collect the remarks in \smallskip {\bf Lemma 2.1.}\ Let $s\in (0,1],$ $q\in (0,\infty],$ $1/s=1+1/q.$ For $a:=(a_k)\in l_1$ and $b:=(b_k)\in l_q$ we have \begin{equation} (\sum_{k=1}^\infty |a_k b_k|^s)^{1/s} \le \sum_{k=1}^\infty |a_k| \cdot (\sum_{k=1}^\infty |b_k|^q)^{1/q}. \label {(1)} %!!! zamenil (2) na (2.1) i nizhe raz \end{equation} Moreover, $$ ||a||_{l_1} = \sup_{||b||_{l_q}=1} (\sum_{k=1}^\infty |a_k b_k|^s)^{1/s} $$ (if $q=\infty$, evident changes must be made in $(1)$). \smallskip {\it Proof}\ of Lemma 2.1. We may consider the case where $q\in (0,\infty).$ Putting $p:= 1/s$ (then $1/p'=1-s=s/q$ and $sp'=q)$, we obtain $$ \sum_{k=1}^\infty |a_k b_k|^s \le (\sum_{k=1}^\infty |a_k|^{sp})^{1/p} \cdot (\sum_{k=1}^\infty |b_k|^{sp'})^{1/p'}= (\sum_{k=1}^\infty |a_k|)^s \cdot (\sum_{k=1}^\infty |b_k|^q)^{s/q}. $$ For the second part: Let $a=(a_k)\in l_1.$ Take $b_k:= \frac{|a_k|^{1/q}}{||a||_{l_1}^{1/q}}.$ Then $\sum_{k=1}^\infty |b_k|^{q}= \sum_{k=1}^\infty \frac{|a_k|}{||a||_{l_1}}=1$ and $$ (\sum_{k=1}^\infty |a_k b_k|^s)^{1/s} = (\sum_{k=1}^\infty \frac{|a_k|^{s/q}}{||a||_{l_1}^{s/q}}\, |a_k|^s)^{1/s}= (\sum_{k=1}^\infty \frac{|a_k|^{s/q+s}}{||a||_{l_1}^{s/q}})^{1/s} $$ $$ = (\sum_{k=1}^\infty \frac{|a_k|^{s(1+1/q)}}{||a||_{l_1}^{s/q}})^{1/s}= (\sum_{k=1}^\infty \frac{|a_k|}{||a||_{l_1}^{s/q}})^{1/s}= \frac {(\sum_{k=1}^\infty |a_k|)^{1/s}}{||a||_{l_1}^{1/q}}= (\sum_{k=1}^\infty |a_k|)^{1/s-1/q}= ||a||_{l_1}. $$ \smallskip {\it Proof}\ of Theorem 2.1 (continuation). $2) \implies 1).$ Suppose that $X$ does not have the $\widetilde{AP_s},$ $1/s=1+1/q.$ Then there is a sequence $(x_n)\in l_q(X)$ (if $q=\infty,$ we consider a sequence from $c_0(X)=l_\infty^0(X)$) such that there exists an $\varepsilon>0$ with the property that for any finite rank operator $R\in X^*\otimes X$ the inequality $\sup_n ||Rx_n-x_n||>\varepsilon$ is valid. Consider the space $C_0(K; X)$ for $K:= \{x_n\}_{n=1}^\infty \cup \{0\}.$ Every operator $U$ in $X$ can be considered as a continuous function on $K$ with values in $X$ by setting $f_U(k):= U(k)$ for $k\in K.$ In particular, for the identity map $\operatorname{id}$ in $X$ and for any $R\in X^*\otimes X$ we have $$ ||f_{\operatorname{id}}-f_R||_{C_0(K;X)}\ge\varepsilon. $$ The subset $\mathcal R:=\overline{\{f_R:\ R\in X^*\otimes X\}}^{C_0(K;X)}$ of $C_0(K; X)$ is a closed linear subspace in $C_0(K; X).$ So, there exists an $X^*$-valued measure $\mu=(x'_k)_{k=1}^\infty\in C_0^*(K; X)= l_1(\{x_n\}_{n=1}^\infty)\cup \{0\}; X)$ such that $\mu|_{\mathcal R}=0$ and $\mu(f_{id})=1.$ In other words, we can find a sequence $(x'_k)$ with $\sum_{k=1}^\infty ||x'_k||<\infty$ such that $\sum_{k=1}^\infty \langle x'_k, x_k\rangle =1$ and $\sum_{k=1}^\infty \langle x'_k, Rx_k\rangle =0$ for any $R\in X^*\otimes X.$ Define a tensor element $z\in X^*\widehat\otimes X$ by $z:= \sum_{k=1}^\infty x'_k\otimes x_k.$ Since $(x_k)\in l_q(X)$ and $(x'_k)\in l_1(X^*),$ we get from Lemma 2.1 that $$ (\sum_{k=1}^\infty ||x'_k||^s\, ||x_k||^s)^{1/s} \le \sum_{k=1}^\infty ||x'_k||\cdot (\sum_{k=1}^\infty ||x_k||^q)^{1/q}. $$ Therefore, $z\in X^*\widehat\otimes_s X,$ $\operatorname{trace}\, z=\sum_{k=1}^\infty \langle x'_k, x_k\rangle =1$ and $\operatorname{trace}\, R\circ z=0$ for every $R\in X^*\otimes X.$ This means that $X$ does not have the $AP_s.$ \smallskip After Theorem 2.1 has been proved, we can make a conclusion: $AP_s= \widetilde{AP}_s$ for any $s\in (0,1].$ Let us mention that this equality appeared firstly (without proofs) in [22, Lemma 2.1]. \vskip 0.23cm {\bf III.}\ Now we are going to discuss some questions around the properties $\widetilde{AP}_{s,\infty}$ and $AP_{s,\infty}.$ The $\widetilde{AP}_{s,\infty}$ was defined above. Recall the definition of the $AP_{s,\infty}$ from, e.g., [23]: We say that a Banach space $X$ has the $AP_{s,\infty},$ $00$ there exists a finite rank operator $R\in X^*\otimes X$ such that $\sup_n ||Rx_n-x_n||<\varepsilon.$ Since $AP_{(s,1)}= AP_{1; q,\infty},$ it is enough to show that if $Y$ is a Banach space, $z\in Y^*\widehat\otimes_{1; q,\infty} X$ and $z\neq0,$ then the corresponding operator $\widetilde z: Y\to X$ is not zero too. Let $z=\sum_{k=1}^\infty a_kb_k\, y'_k\otimes x_k$ be a representation of $z$ with $(x_k), (y'_k)$ bounded, $(a_k)\in l_1,$ $(b_k)\in l^0_{q\infty}$ and $b_k\searrow0.$ Then $(\widetilde x_k:=b_kx_k)\in l^0_{q \infty}$ and, for an $\varepsilon>0$ small enough (to be chosen), we can find an operator $R\in X^*\otimes X$ with the property that $\sup_n ||R\widetilde x_n-\widetilde x_n||\le \varepsilon.$ Since $z\neq0,$ we can find an operator $V\in L(Y^*, X^*)$ such that $\sum_{k=1}^\infty a_k\, \langle Vy'_k, \widetilde x_k\rangle =1.$ Now, when $V$ is chosen, we have $$ 1= \sum_{k=1}^\infty a_k\, \langle Vy'_k, \widetilde x_k-R\widetilde x_k\rangle +\sum_{k=1}^\infty a_k\, \langle Vy'_k, R\widetilde x_k\rangle $$ $$\le \varepsilon\, ||(a_k)||_{l_1}\, ||V||\cdot const + |\sum_{k=1}^\infty a_kb_k\, \langle R^*Vy'_k, x_k\rangle |, $$ and, if $\varepsilon$ is small enough, we get for the finite rank operator $R^*V: Y^*\to X^*$ that $$ |\operatorname{trace}\, z^t\circ (R^*V)|= |\operatorname{trace}\, (R^*V)\circ z^t|= |\sum_{k=1}^\infty a_kb_k\, \langle R^*Vy'_k, x_k\rangle |>0. $$ The last sum is the nuclear trace of the tensor element $\sum_{k=1}^\infty a_kb_k\, R^*Vy'_k\otimes x_k,$ which is a composition $R\circ z_0$ of the finite rank operator $R$ and the tensor element $\sum_{k=1}^\infty a_kb_k\, Vy'_k\otimes x_k,$ that belongs to the tensor product $X^*\widehat\otimes_{1;q,\infty} X.$ It follows that both $z_0$ and $z$ generate the non-zero operators $\widetilde z_0$ and $\widetilde z.$ \smallskip {\it Remark} 3.4.\ Because of the equality $\widehat\otimes_{1;q,\infty}= \widehat\otimes_{(s,1)},$ it follows from the proof of Theorem 3.3 that $X$ has the $AP_{(s,1)}$ iff the canonical mapping $X^*\widehat\otimes_{(s,1)}X \to L(X)$ is one-to-one (just like in the case of the classical Grothendieck approximation property). \smallskip {\it Remark} 3.5.\ Of course, it follows from Theorem 3.3 that every Banach space has the $AP_{(2/3,1)},$ but it is trivial because of the implication $$ AP^0_{(2/3,\infty)} \equiv AP_{2/3,\infty} \implies AP_{(2/3,w)} \ \text{for any } w<\infty $$ (and, again, since every $X$ has the $AP_{2/3,\infty}!).$ \smallskip Our question in Remark 3.3 can be reformulated now as: $({}^*)$\ Is it true that the $AP_{(s,1)}$ implies the $AP_{s,\infty}?$ \vskip 0.23cm {\bf IV.}\ Let us consider an application of the previous considerations. Now we know, in particular, that every Banach space has the $AP_{(2/3,1)}.$ On the other hand, the corresponding operator ideal $N_{(2/3,1)}$ (related to the Lorentz space $l_{2/3\, 1 })$ has the eigenvalue type $l_{1}$ (see, e.g., [8, p. 243]). Since the continuous trace is unique on $\widehat\otimes_{(2/3,1)}$ and $\widehat\otimes_{(2/3, 1)}= N_{(2/3, 1)},$ it follows from White's results [32] that for each Banach space $X$ and for every operator $T\in N_{(2/3,1)}(X,X)$ the (nuclear) trace of $T$ is well defined and equals the sum of all eigenvalues of $T:$ $$ \operatorname{trace}\, T= \sum_{k=1}^\infty \mu_k(T)\ \text{(eigenvalues)}\, \forall\, X,\ \forall\, T\in N_{(2/3,1)}(X) $$ (on the right is the so-called "spectral sum" of $T).$ More precisely, the last statement follows from Theorem 4.1 below. Let us explain in more details how we apply a result of M.C. White. To do this, we formulate and prove a theorem which is almost an immediate consequence of White's theorem. \smallskip {\bf Theorem 4.1.}\ Let $A$ be a quasi-Banach operator ideal, $X$ be a Banach space, for which the set of all finite rank operators is dense in the space $A(X).$ Suppose that the natural functional "$\operatorname{trace}\,$" is bounded on the subspace of all finite rank operators of $A(X)$ (and, therefore, can be extended to a continuous functional "$\operatorname{trace}_A$" on the whole space $A(X)$). If the quasi-Banach operator ideal $A$ is of eigenvalue type $l_1,$ then the spectral trace (= "spectral sum") is continuous on the space $A(X)$ and for any operator $T\in A(X)$ we have $$ \operatorname{trace}_A(T) = \sum_{n=1}^{\infty} \mu_n(T). $$ where $(\mu_n(T))_{n=1}^\infty$ is the sequence of all eigenvalues of $T$ counted according to their multiplicities. \smallskip {\it Proof}\ of Theorem 4.1. \ Let $T\in A(X).$ By the assumption, the sequence $\{\mu_n(T)\}_{n=1}^\infty$ of all eigenvalues of $T$ (counted according to their multiplicities) is in $l_1.$ Since the quasi-normed ideal $A$ is of spectral (= eigenvalue) type $l_1,$ we can apply the main result from the paper [32] of M.C. White, which asserts: %that: $({}^*{}_*{}^*)$\, {\it If $J$ is a quasi-Banach operator ideal with eigenvalue type $l_1,$ then the spectral sum is a trace on the ideal $J$}. Recall (see [18], 6.5.1.1, or Definition 2.1 in [32]) that a {\it trace}\ on an operator ideal $J$ is a class of complex-valued functions, all of which they write as $\tau,$ one for each component $J(E,E)$ (where $E$ is a Banach space) so that (i)\ $\tau(e'\otimes e)= \langle e',e\rangle$ for all $e'\in E^*, e\in E;$ (ii)\ $\tau(AU)=\tau(UA)$ for all Banach spaces $F$ and operators $U\in J(E,F)$ and $A\in L(F,E); $ (iii)\ $\tau(S+U)=\tau(S) +\tau(U)$ for all $S,U\in J(E,E);$ (iv)\ $\tau(\lambda U)= \lambda \tau(U)$ for all $\lambda\in \Bbb C$ and $U\in J(E,E).$ Our operator $T$ belongs to the space $A(X,X)=A(X)$ and $A$ is of eigenvalue type $l_1.$ Thus, the assertion $({}^*{}_*{}^*)$ implies that the spectral sum $\mu$ defined by $\mu(U):= \sum_{n=1}^\infty \mu_n(U)$ for $U\in A(E,E)$ is a trace on $A.$ By the principle of uniform boundedness (see [17], 3.4.6 (page 152), or [15]), there exists a constant $C>0$ such that $$ |\mu(U)|\le ||\{\mu_n(U)\}||_{l_1} \le C\, a(U) $$ for all Banach spaces $E$ and operators $U\in A(E,E). $ Now, remembering that all operators in $A(X)$ can be approximated by finite rank operators and taking in account the conditions (iii)--(iv) for $\tau=\mu$, we obtain that the $A$-trace, i.e. $\operatorname{trace}_A T,$ of our operator $T$ coincides with $\mu (T)$ (recall that the continuous trace is uniquely defined in such a situation, that is on the space $A(X);$ cf. [18], 6.5.1.2). \smallskip Since $\widehat\otimes_{1;2,\infty}= \widehat\otimes_{(2/3,1)}$ (see Theorem 3.3), we can reformulate the result, which we considered in the very beginning of this section, as \smallskip {\bf Corollary 4.1.}\ For each Banach space $X$ and for every operator $T\in N_{1;2,\infty}(X,X)$ the (nuclear) trace of $T$ is well defined and equals the sum of all eigenvalues of $T:$ $$ \operatorname{trace}\, T= \sum_{k=1}^\infty \mu_k(T)\ \text{(eigenvalues)}\, \forall\, X,\ \forall\, T\in N_{(1;2,\infty)}(X). $$ \smallskip {\it Remark \rm4.1}:\, Recall that A. Grothendieck [7] has obtained the last assertion for the case of $2/3$-nuclear operators, i.e. for the ideal $N_{2/3}= N_{(2/3,\, 2/3)}$ (note that $l_{2/3}\subset l_{2/3\, 1}).$ % \smallskip \vskip 0.23cm {\bf V.}\ The discussion in Section I shows that, for $p\in[1,\infty],$ any subspace of any quotient (= any quotient of any subspace) of an $L_p$-space possesses the $\widetilde{AP}_s$ (even the $\widetilde{AP}_{s,\infty})$ with $1/s=1+|1/2-1/p|.$ We apply now these facts together with White's theorem for proving some more theorems concerning the distributions of eigenvalues of the nuclear operators. Below we will use Theorem 2.1 and, therefore, the fact that any subspace of any quotient of an $L_p$-space possesses the ${AP}_s$ (where $p,s$ as above). Thus, for such Banach spaces $X,$ we can identify the tensor product $X^*\widehat\otimes_s X$ with its canonical image in the space $L(X)=L(X,X),$ i. e., with the space $N_s(X)$ of all $s$-nuclear operators in $X,$ equipped with the quasi-norm induced from $X^*\widehat\otimes_s X.$ We give below relatively simple proofs of some recent results from the papers [25] and [28]. \smallskip {\bf Theorem 5.1.}\ Let $X$ be a subspace of an $L_p$-space, $1\le p\le \infty.$ If $T\in N_s(X,X),$\, where $1/s=1+|1/2-1/p|,$ \, then 1.\, the (nuclear) trace of $T$ is well defined, 2.\, $\sum_{n=1}^\infty |\mu_n(T)|<\infty,$ where $\{\mu_n(T)\}$ is the system of all eigenvalues of the operator $T$ (written in according to their algebraic multiplicities) and $$ \operatorname{trace}\, T= \sum_{n=1}^\infty \mu_n(T). $$ \vskip 0.2cm {\it Proof}.\ Let $X$ be a subspace of an $L_p$-space $L_p(\mu)$ and $T\in N_s(X,X)$ with an s-nuclear representation $$ T=\sum_{k=1}^\infty \lambda_k x'_k\otimes x_k, $$ where $||x'_k||, ||x_k||=1$ and $\lambda_k\ge 0,$ $\sum_{k=1}^\infty \lambda_k^s<\infty.$ By Hahn-Banach, we can find the functionals $\widetilde x'_k\in L^*_p(\mu)$ \, $(k=1,2,\dots)$ with the same norms as for the corresponding functionals $x'_k$ and so that $\widetilde x'_k|_X=x'_k$ for every $k.$ Denote by $\widetilde T$ the operator $$ \widetilde T: L_p(\mu)\to X,\ \widetilde T:= \sum_{k=1}^\infty \lambda_k \widetilde x'_k\otimes x_k, $$ and by $j: X\to L_p(\mu)$ the natural injection. Since the space $X$ has the property $AP_s,$ we have $N_s(L_p(\mu), X)= L^*_p(\mu)\widehat\otimes_s X$ and, therefore, the nuclear traces of the operators $j\widetilde T$ and $\widetilde Tj$ are well defined. We have a diagram $$ X\overset{j}\to L_p(\mu) \overset{\widetilde T}\to X \overset{j}\to L_p(\mu), %ooo!!! ovs check.......... i nizhe $$ in which $\widetilde Tj = T\in N_s(X).$ Hence, the complete systems of eigenvalues of the operators $T=\widetilde Tj $ and $j\widetilde T\in N_s(L_p(\mu))$ coincide. Applying Theorem 2.b.13 from [10] (see also [25]), we obtain that the sequence $(\mu_k(j\widetilde T))$ is summable. Therefore, we have $\mu_k(T)\in l_1$ and we can apply Theorem 4.1. But we apply the theorem firstly for the simplest case (later on we will continue the proof of our theorem 5.1). \smallskip The first assertion of the next theorem is due to A. Grothendieck [7], %!!!R the second one was proved by H. K\"onig in [11]. Surprisingly, %!!!R but we could not find anywhere the main statement of the theorem about coincidence of the nuclear and spectral traces, neither in the monographs, nor in the mathematical journals. So we have no reference for this statement and have to formulate and to prove the next theorem here. Let us remark that, in any case, this theorem was proved (as a partial case of the proved there our Theorem 5.1) in [25]. %!!!R \smallskip {\bf Theorem $\mathbf5.1'.$}\ Let $L$ be an $L_p$-space, $1\le p\le \infty.$ If $T\in N_s(L, L),$\, where $1/s=1+|1/2-1/p|,$ \, then 1.\, the (nuclear) trace of $T$ is well defined, 2.\, $\sum_{n=1}^\infty |\mu_n(T)|<\infty,$ where $\{\mu_n(T)\}$ is the system of all eigenvalues of the operator $T$ (written in according to their algebraic multiplicities) and $$ \operatorname{trace}\, T= \sum_{n=1}^\infty \mu_n(T). $$ \vskip 0.2cm {\it Proof}.\ As we have said above, the assertions 1 and 2 are well known. To prove the last equality, consider the Banach operator ideal $\mathcal L_p$ of all operators which can be factored through $L_p$-spaces. Then the product $\mathcal L_p\circ N_s$ is a quasi-Banach operator ideal of spectral (=eigenvalue) type $l_1$ (e.g., by the assertion 2, proved earlier by H. K\"onig [11]). %!!!R Now it is enough to apply Theorem 4.1 to finish the proof. \smallskip {\it Proof \, of Theorem }5.1 \, (continuation). As we have said, the complete systems of eigenvalues of the operators $T=\widetilde Tj $ and $j\widetilde T\in N_s(L_p(\mu))$ coincide. By Theorem 5.1', $$ \operatorname{trace}\, j\widetilde T = \sum_{k=1}^\infty \lambda_k \, \langle \widetilde x'_k, jx_k\rangle = \sum_{n=1}^\infty \mu_n(j\widetilde T), $$ the last sum is equal to $$ \sum_{n=1}^\infty \mu_n(T) $$ and the sum in the middle is $$ \sum_{k=1}^\infty \lambda_k \, \langle \widetilde x'_k, jx_k\rangle = \sum_{k=1}^\infty \lambda_k\, \langle x'_k, x_k\rangle =\operatorname{trace}\, T. $$ The (nuclear) trace of the operator $T$ is well defined, because the space $X$ has the $AP_s.$ Therefore, $$ \operatorname{trace}\, T= \sum_{n=1}^\infty \mu_n(T), $$ and we are done. \smallskip If $Y$ is a quotient of an $L_p$-space, then, for a compact operator %!!! $U\in L(E,E),$ the adjoint $U^*$ is also a compact operator and these two operators have the same eigenvalues $\mu\neq0$ with the same multiplicities (see, e.g., [17], Theorem 3.2.26, or [5], Exercise VII.5.35). %%!!!R from MNwe Also, any quotient of an $L_p$-space has the $AP_s$ (where $p,s$ are as above). So, it follows immediately from the just proved Theorem 5.1 \smallskip {\bf Corollary 5.1.}\ Let $Y$ be a quotient of an $L_p$-space, $1\le p\le \infty.$ If $T\in N_s(Y,Y),$\, where $1/s=1+|1/2-1/p|,$ \, then 1.\, the (nuclear) trace of $T$ is well defined, 2.\, $\sum_{n=1}^\infty |\mu_n(T)|<\infty,$ where $\{\mu_n(T)\}$ is the system of all eigenvalues of the operator $T$ (written in according to their algebraic multiplicities) and $$ \operatorname{trace}\, T= \sum_{n=1}^\infty \mu_n(T). $$ \vskip 0.2cm We used above some facts from Section I. After Theorem 5.1 and its consequence have been proved, we are ready to present a simple prove of the corresponding result on the subspaces of quotients of the $L_p$-spaces (recall that, again, all such spaces have the $AP_s$ with $s$ and $p$ satisfying the same conditions). \smallskip {\bf Theorem 5.2.}\ Let $W$ be a quotient of a subspace (= a subspace of a quotient) of an $L_p$-space, $1\le p\le \infty.$ If $T\in N_s(W,W),$\, where $1/s=1+|1/2-1/p|,$ \, then 1.\, the (nuclear) trace of $T$ is well defined, 2.\, $\sum_{n=1}^\infty |\mu_n(T)|<\infty,$ where $\{\mu_n(T)\}$ is the system of all eigenvalues of the operator $T$ (written in according to their algebraic multiplicities) and $$ \operatorname{trace}\, T= \sum_{n=1}^\infty \mu_n(T). $$ \vskip 0.2cm {\it Proof}.\ Let $L_p(\mu)$ be an $L_p$-space. Take Banach subspaces $X_0\subset X\subset L_p(\mu)$ and consider the quotient $X/X_0.$ If $T\in N_s(X/X_0, X/X_0)$ (=$(X/X_0)^*\widehat\otimes_s X/X_0),$ then $T$ admits a factorization of the type $$ X/X_0 \overset{A}\to c_0 \overset{D}\to l_1 \overset{B}\to X/X_0, %ooo $$ where $A,B$ are continuous and $D$ is a diagonal operator with a diagonal from $l_s.$ Denoting by $\varphi: X\to X/X_0$ the factor map from $X$ onto $X/X_0$ and taking a lifting $\Phi: l_1\to X$ for $B$ with $B=\varphi\Phi,$ we obtain that the maps $\varphi \Phi DA: X/X_0\to X/X_0$ and $\Phi DA\varphi: X\to X$ have the same eigenvalues $\mu\neq0$ with the same multiplicities: $$ X\overset{\varphi}\to X/X_0 \overset{A}\to c_0 \overset{D}\to l_1 \overset{\Phi}\to X \overset{\varphi}\to X/X_0, %ooo $$ The spaces $X$ and $X/X_0$ have the $AP_s.$ Therefore, we have (cf. the proof of Theorem 5.1) $$ \operatorname{trace}\, \varphi \Phi DA = \operatorname{trace}\, \Phi DA\varphi. $$ Since $X$ is a subspace of $L_p(\mu),$ we have, by Theorem 5.1, $$ \operatorname{trace}\, \Phi DA\varphi= \sum_{n=1}^\infty \mu_n(\Phi DA\varphi). $$ Therefore, $$ \operatorname{trace}\, T = \operatorname{trace}\, BDA = \operatorname{trace}\, \varphi \Phi DA = \sum_{n=1}^\infty \mu_n(\Phi DA\varphi) $$ $$ = \sum_{n=1}^\infty \mu_n(\varphi\Phi DA) = \sum_{n=1}^\infty \mu_n(BDA) =\operatorname{trace}\, T. $$ % \smallskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip 0.23cm {\bf VI.}\ As is well known, in the classical case of the Grothendieck approximation property $AP,$ if $X^*$ has the $AP$ then the space $X$ also has this property. We will show now that the same is true for all approximation properties which are under consideration in this paper. Denote by $\widehat\otimes_\alpha$ any of the tensor product $\widehat\otimes_s,$ $\widehat\otimes_{s,\infty},$ $\widehat\otimes_{t; p,r}$, $\widehat\otimes_{(r,w)}$ with the parameters (see above), for which all these tensor products are the linear subspaces of the projective tensor product $\widehat\otimes.$ Also, let us say that a Banach space $X$ has the $AP_\alpha,$ if it is possesses the corresponding approximation property (i.e., $AP_s,$ $AP_{s,\infty}$ etc.). \smallskip We need the following auxiliary result which may be of its own interest (compare with Remark 3.4). \smallskip {\bf Proposition 6.1} A Banach space $X$ has the $AP_\alpha$ iff the canonical map $X^*\widehat\otimes_\alpha X\to L(X)$ is one-to-one. \smallskip {\it Proof}.\ Suppose that the canonical map $X^*\widehat\otimes_\alpha X\to L(X)$ is one-to-one, but there exists a Banach space $Y$ such that the natural map $Y^*\widehat\otimes_\alpha X\to L(Y, X)$ is not injective. Let $z\in Y^*\widehat\otimes_\alpha X$ be such that $z\neq0$ and the associated operator $\widetilde z$ is a 0-operator. Then we can find an operator $V$ from $L(Y^*,X^*)$ (the dual space to the projective tensor product $Y^*\widehat\otimes X)$ so that $\operatorname{trace}\, V\circ z^t=1,$ where, as usual, $z^t$ is the transposed tensor element, $z*t\in X\widehat\otimes Y^*.$ Since $V\circ z^t\in X\widehat\otimes X^*$ and $\operatorname{trace}\, V\circ z^t=1,$ the tensor element $(V\circ z^t)^t$ (which, evidently, belongs to $X^*\widehat\otimes_\alpha X)$ is not zero. On the other hand, the operator induced by this element must be a 0-operator. Contradiction. \smallskip %\noindent {\bf Proposition 6.2.}\ With the above understanding, if the dual space $Y^*$ has the $AP_\alpha,$ then $Y$ has the $AP_\alpha$ too. \medskip %\noindent {\it Proof}.\, We use Proposition 6.1. As it is known [7], the projective tensor product $Y^*\widehat\otimes Y$ is a Banach subspace of the tensor product $Y^*\widehat\otimes Y^{**}.$ The tensor product $Y^*\widehat\otimes_\alpha Y $ is a linear subspace of $Y^*\widehat\otimes Y,$ as well as $Y^*\widehat\otimes_\alpha Y^{**}$ is a linear subspace of $Y^*\widehat\otimes Y^{**}.$ Therefore, the natural map $Y^*\widehat\otimes_\alpha Y \to Y^*\widehat\otimes_\alpha Y^{**} $ is one-to-one. Now if $Y^*$ has the $AP_\alpha,$ then the canonical map $Y^{**}\widehat\otimes_\alpha Y^* \to L(Y^*,Y^*)$ is one-to-one. Since we can identify the tensor product $Y^{**}\widehat\otimes_\alpha Y^*$ with the tensor product $Y^{*}\widehat\otimes_\alpha Y^{**}$ (because of the "symmetries" in the definitions of the corresponding tensor products), it follows that the natural map $Y^*\widehat\otimes_\alpha Y \to L(Y,Y)$ is one-to-one. Thus, if $Y^*$ has the $AP_\alpha,$ then $Y$ has the $AP_\alpha$ too. \smallskip % We will use Lemma 4 in the proof of Theorem 1 in the next section. % \smallskip %%%%%%%% Zamechanie o tochnosti - perepravit' !!! % \noindent {\it Remark \rm6.1}:\ The inverse statement is not true. For example, %!!! rem No... if $s\in(2/3,1],$ then there exists a Banach space, possessing the Grothendieck approximation property, whose dual does not have the $AP_s$ (it is well known for the case where $s=1).$ Moreover, if $s\in(2/3,1],$ then we can find a Banach space $W$ such that $W$ has a Schauder basis and $W^*$ does not have the $AP_s.$ Indeed, let $E$ be a separable reflexive Banach space without the $AP_s$ (see [21] or [23]). %!!! Ref me or me!!! Let $ Z$ be a separable space such that $ Z^{**}$ has a basis and there exists a linear homomorphism $ \varphi$ from $ Z^{**}$ onto $ E^*$ %with the kernel %$ Z\subset Z^{**}$ so that the subspace $ \varphi^*(E)$ is complemented in $ Z^{***}$ and, moreover, $Z^{***}\cong \varphi^*(E)\oplus Z^*$ (see [13, Proof of Corollary 1]). Put $W:= Z^{**}.$ This (second dual) space $W$ has a Schauder basis and its dual $W^*$ does not have the $AP_s.$ % \smallskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip 0.23cm {\bf VII.}\ Let us consider some more notions of the approximation properties associated with some other tensor products. For Banach spaces $X$ and $Y$ and $r\in (0,1], p\in [1,2],$ define a quasi-norm $|| \cdot||_{N_{[r,p]}}$ on the tensor product $X^*\otimes Y$ by \begin{equation}\nonumber %!!! 888 nuzhna numeracija \|u\|_{N_{[r,p]}}:=\inf\left\{\|(x_{i}')_{i=1}^{n}\|_{\ell_{r}(X^*)}\cdot\|(y_{i})_{i=1}^{n} \|_{\ell_{p'}^{w}(Y)}:\ u=\sum_{i=1}^{n}x_{i}'\otimes y_{i}\right\} \label {(3)} \end{equation} Here we denote, as usual, by $l_r(X^*)$ and $l_q^w(Y)$ the spaces of $r$-absolutely summable and weakly $q$-summable sequences, respectively. Denote by $X^{*}\widehat\otimes_{[r,p]}Y$ the completion of the space $\left(X^{*}\otimes Y,\ \|\cdot\|_{N_{[r,p]}}\right).$ We have a natural continuous injection $$j_{[r,p]}: X^{*}\widehat\otimes_{[r,p]}Y\rightarrow X^{*}\widehat\otimes Y$$ with $||j_{[r,p]}||\le1.$ Every element $u\in X^{*}\widehat\otimes_{[r,p]}Y$ has a representation of the type $u=\sum_{i=1}^{\infty}x_{i}'\otimes y_{i},$ where $(x_{i}')_{i=1}^{\infty}\in\ell_{r}(X^{*})$ and $(y_{i})_{i=1}^{\infty}\in \ell_{p'}^{w}(Y)$. Consider the natural mappings $$ X^{*}\widehat\otimes_{[r,p]} Y \overset{{j}_{[r,p]}}\to X^{*}\widehat\otimes Y \overset{{j }}\to L(X,Y). $$ The image of the tensor product $X^{*}\widehat\otimes_{[r,p]} Y $ under the composition $\widetilde{j}_{[r,p]}:={j}\circ j_{[r,p]}$ is denoted by $N_{[r,p]}(X,Y)$. This is a quasi-Banach space of the $(r,p)$-nuclear operators (the quasi-norm is induced from the tensor product $X^{*}\widehat\otimes_{[r,p]} Y).$ It is not difficult to see that every operator $T\in N_{[r,p]}(X,Y)$ admits a factorization of the kind % $$ \begin{equation} X \overset{A}\to c_0 \overset{D_r}\to l_1 \overset{i}\to l_p \overset{B}\to Y, \label {(3)} %ooo \end{equation} % $$ where $A,B$ are compact, $i$ is the injection, $D_r$ is a diagonal operator with a diagonal from $l_r.$ Also, every operator, which can be factored in such a way, is in $N_{[r,p]}(X,Y).$ \smallskip By the analogous way, we define the tensor product $X^{*}\widehat\otimes^{[r,p]}Y$ and the quasi-normed operator ideals $N^{[r,p]}(X,Y).$ Namely, $X^{*}\widehat\otimes^{[r,p]}Y$ is a linear subspace of the projective tensor product $X^*\widehat\otimes Y,$ consisting of tensor elements $z$ which admit a representation $$ u=\sum_{i=1}^{\infty}x_{i}'\otimes y_{i}, $$ where $(x_{i}')_{i=1}^{\infty}\in\ell_{p'}^w(X^{*})$ and $(x_{i})_{i=1}^{\infty}\in \ell_{r}(Y).$ Its canonical image in $L(X,Y)$ is the quasi-normed space $N^{[r,p]}(X,Y).$ It is not difficult to see that every operator $T\in N^{[r,p]}(X,Y)$ admits a factorization of the kind $$ X \overset{A}\to l_{p'} \overset{D_r}\to c_0 \overset{i}\to l_1 \overset{B}\to Y, %ooo $$ where $A,B$ are compact, $i$ is the injection, $D_r$ is a diagonal operator with a diagonal from $l_r.$ Also, every operator, which can be factored in such a way, is in $N^{[r,p]}(X,Y).$ It is clear that $T\in N_{[r,p]}(X, Y)$ implies $T^*\in N^{[r,p]}(Y^*, X^*)$ and $T\in N^{[r,p]}(X,Y)$ implies $T^*\in N_{[r,p]}(Y^*, X^*).$ Inverse is not true (see, e.g., Example 8.3 below). %!!! Check!!! Now we can define the notions of the corresponding approximation properties by the usual way. We say that he space $X$ has the $AP_{[r,p]}$ (respectively, the $AP^{[r,p]})$ if for every Banach space $Y$ the natural mapping $ Y^{*}\widehat\otimes_{[r,p]} X\to L(Y,X)$ (respectively, $ Y^{*}\widehat\otimes^{[r,p]} X\to L(Y,X)$) is one-to-one. It can be seen that a Banach space $X$ has the $AP_{[r,p]}$ (or $AP^{[r,p]})$ iff the canonical map $X^*\widehat\otimes_{[r,p]} X\to L(X)$ (or $X^*\widehat\otimes^{[r,p]} X\to L(X)$) is one-to-one (the proof is essentially the same as the proof of Theorem 6.1). Also, if $X^*$ has the $AP_{[r,p]}$ (or $AP^{[r,p]})$ then $X$ has the $AP^{[r,p]}$ (or $AP_{[r,p]})$ (the proof is the same as in Theorem 6.2). \smallskip {\bf Theorem 7.1.}\ Let $1/r-1/p=1/2.$ Every Banach space has the properties $AP_{[r,p]}$ and $AP^{[r,p]}.$ \smallskip {\it Proof}.\ Suppose that $X\notin AP_{[r,p]}$ where $1/r-1/p=1/2.$ Let $z\in X^*\widehat\otimes_{[r,p]} X$ be an element such that $\operatorname{trace}\, z=1, \tilde z=0.$ Since $z=\sum x'_k\otimes x_k,$ where $(x'_k)\in l_r(X^*) $ and $ (x_k) $ is weakly $ p'$-summable, the operator $ \tilde z $ can be factored as $$ \tilde z:\ X\overset{A}\to l_\infty \overset{\Delta}\to l_1 \overset{j}\to l_p \overset{V}\to X, $$ where all the operators are continuous, $ j $ is an injection, $ \Delta $ is a diagonal operator with a diagonal from $ l_r. $ Since $\tilde z=0,$ we have $V|_{j\Delta A(X)}=0.$ Consider $S:= j\Delta AV: l_p\to l_p.$ Evidently, $S^2=0$ and $\operatorname{trace}\, S=\operatorname{trace}\, z=1.$ Since $S\in N_r(l_p,l_p),$ its nuclear trace equals the sum of all its eigenvalues (see Theorem $\mathrm5.1'$ above). This contradicts the fact that $S^2=0.$ Now, let $Y$ be another Banach space and put $X:= Y^*.$ We have shown that $X$ has the $AP_{[r,p]}.$ Therefore (see remarks before the formulation of Theorem 7.1), $Y$ has the $AP^{[r,p]}.$ \vskip0.1cm %%% We are ready to apply the above results to the investigation of eigenvalues problems for $N_{[r,p]}$- and $N^{[r,p]}$-operators. The first theorem below was proved in [26] by using Fredholm Theory. The same proof can be applied for the second theorem. Below we present very different simple proofs of them. \smallskip {\bf Theorem 7.2.}\ Let $1/r-1/p=1/2.$ For every Banach space $X$ and every operator $T\in N_{[r,p]}(X)$, $\operatorname{trace}\,T$ is well defined and if $(\mu_{i})_{i=1}^{\infty}$ is a system of all eigenvalues of $T,$ then $(\mu_{i})_{i=1}^{\infty}\in l_1$ and \begin{equation}\nonumber \operatorname{trace}\,T=\sum_{i=1}^{\infty}\mu_{i}. \end{equation} \smallskip {\bf Theorem 7.3.}\ Let $1/r-1/p=1/2.$ For every Banach space $X$ and every operator $T\in N^{[r,p]}(X)$, $\operatorname{trace}\,T$ is well defined and if $(\mu_{i})_{i=1}^{\infty}$ is a system of all eigenvalues of $T,$ then $(\mu_{i})_{i=1}^{\infty}\in l_1$ and \begin{equation}\nonumber \operatorname{trace}\,T=\sum_{i=1}^{\infty}\mu_{i}. \end{equation} \smallskip Both theorems can be proved by the analogues methods and the proofs are almost the same as the proof of Theorem 5.2 (by using Theorem 7.1). %So we omit it here. On the other hand, Theorem 7.3 is a consequence of Theorem 7.2 and vice versa. Let us give, firstly, a simple proof of Theorem 7.2 and then deduce Theorem 7.3. {\it Proof}.\, Suppose that $T\in N_{[r,p]}(X)$ and consider a factorization (3) (in which $Y=X)$ of the operator $T.$ The sequence $(\mu_k)$ of all eigenvalues of $T$ is the same as the sequence of all eigenvalues of the operator $iD_rAB,$ which maps $l_p$ into $l_p.$ The last operator is $r$-nuclear, where $p\in [1,2]$ and $1/r=1/p+1/2.$ By Theorem $\mathrm5.1',$ $(\mu_k)\in l_1$ and $\operatorname{trace}\, iD_rAB= \sum \mu_k.$ Since the trace of $T$ is well defined (Theorem 7.1), it is clear that $\operatorname{trace}\, T=\operatorname{trace}\, iD_rAB=\sum \mu_k.$ Now, let $T\in N^{[r,p]}(X).$ Then $T^*\in N_{[r,p]}(X^*)$ and $\operatorname{trace}\, T^*=\operatorname{trace}\, T$ (apply Theorem 7.1). Since the operators $T$ and $T^*$ have the same systems of eigenvalues, Theorem 7.3 follows from just proved statement of Theorem 7.2. \smallskip % {\it Proof}.\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip 0.23cm {\bf VIII.}\ The next examples are taken from [26], where one can find the corresponding proofs. They show that all the above affirmative results concerning approximation properties and trace-formulas are sharp. \smallskip {\bf Example 8.1.}\ Let $r\in(2/3,1], p\in(1,2], 1/r-1/p=1/2.$ %There exist Banach spaces such that for each r There exist Banach spaces $E$ and $V,$ $z_0\in E^*\widehat\otimes V, S\in L(V,E)$ so that for every $p_0\in [1,p)$ 1)\, $z_0\in E^*\widehat\otimes_{[r,1]} V;$ 2)\, $V$ has a basis; 3)\, $V$ is the space of type $p_0$ and of cotype $2;$ 4)\, $S\circ z_0\in E^*\widehat\otimes_{[r,p_0]} E;$ 5)\, $\operatorname{trace}\, S\circ z_0=1;$ 6)\, the corresponding operator $\widetilde{S\circ z_0}$ is a 0-operator and, therefore, has no nonzero eigenvalues. \vskip0.1cm %%%%%%% {\bf Example 8.2.}\ Let $r\in[2/3,1), p\in[1,2), 1/r-1/p=1/2.$ %There exist Banach spaces such that for each r There exist Banach spaces $E$ and $V,$ $z_0\in E^*\widehat\otimes V, S\in L(V,E)$ so that for every $\epsilon>0$ 1)\, $z_0\in E^*\widehat\otimes_{[r+\epsilon,1]} V;$ 2)\, $V$ has a basis; 3)\, $S\circ z_0\in E^*\widehat\otimes_{[r+\epsilon,p]} E;$ 4)\, $\operatorname{trace}\, S\circ z_0=1;$ 5)\, the corresponding operator $\widetilde{S\circ z_0}$ is a 0-operator and therefore, has no nonzero eigenvalues. \vskip0.1cm {\bf Example 8.3.}\ Let $r\in(2/3,1]$, $p\in(1,2], 1/r-1/p=1/2.$ There exist two separable Banach spaces $X$ and $Z$ so that (i)\, $Z^{**}$ has a basis; (ii)\, $\exists \, V\in X^*\widehat\otimes Z^{**}: \ V=\sum_{k=1}^\infty x'_k\otimes z''_k; $\ $(x'_k)$ weakly $p'_0$-summable for each $p_0\in [1,p);$ $(z''_k)\in l_r(Z^{**});$ (iii)\, $V(X)\subset Z; $ the operator $V$ is not nuclear as a map from $X$ into $Z.$ Moreover, there exists an operator $U:Z^{**}\to Z$ such that $(\alpha)$\, $\pi_ZU\in N^{[r,p_0]}(Z^{**},Z^{**})=Z^{***}\widehat\otimes^{[r,p_0]} Z^{**},\ \forall \, p_0\in[1,p);$ $(\beta)$\, $U$ is not nuclear as a map from $Z^{**}$ into $Z;$ $(\gamma)$\, $\operatorname{trace}\, \pi_ZU=1;$ $(\delta)$\, $\pi_ZU: Z^{**}\to Z^{**}$ has no nonzero eigenvalues. \medskip {\bf Acknowledgement.}\, The author would like to thank the referee for helpful remarks and Alexander Alenitsyn for his technical help. \vskip1.1cm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{00} \bibitem{1} J. Bergh, J. L\"ofstr\"om: \textit{ Interpolation spaces}, Berlin, Heidelberg, New York: Springer-Verlag (1976). \bibitem{2} J. Bourgain, O.I. Reinov: \textit{On the approximation properties for the space $H^\infty,$} Math. Nachr. \textbf{122}, No. 2 (1985), 19-27. \bibitem{3} A.M. Davie: \textit{The approximation problem for Banach spaces}, Bull. London Math. Soc. \textbf{5} (1973), 261--266. \bibitem{4} J. Diestel, H. Jarchow, A. Tonge: \textit{Absolutely summing operators}, Cambridge Univ. Press, 1995. \bibitem{5}N. Dunford, J. Schwartz: \textit{Linear operators, Part I}, Interscience, New York (1958). \bibitem{6} P. Enflo: \textit{A counterexample to the approximation property in Banach spaces}, Acta Math. \textbf{130} (1973), 309-317. \bibitem{7} A.~Grothendieck: \textit{Produits tensoriels topologiques et \'espaces nucl\'eaires}, {Mem. Amer. Math. Soc.} \textbf{16}(1955). \bibitem{8} A. Hinrichs, A. Pietsch: $p$-nuclear operators in the sense of Grothendieck, Math. Nachr. \textbf{283}, No. 2 (2010), 232--261. \bibitem{9} M. J. Kadec, M. G. Snobar: \textit{Certain functionals on the Minkowski compactum} (Russian), {Mat. Zametki} \textbf{10} (1971), 453-458. \bibitem{10} H. Konig: \textit{Eigenvalue distribution of compact operators}, Birkh\"auser (1986). \bibitem{11} H. Konig: \textit{On the eigenvalue spectrum of certain operator ideals}, Coll. Math. \textbf{44} (1981), 15--28. \bibitem{12} V.\,B.~Lidski\v{\i}: \textit{Nonselfadjoint operators having a trace} (Russian), {Dokl. Akad. Nauk SSSR} \textbf{125} (1959), 485--487. \bibitem{13} J. Lindenstrauss: \textit{On James' paper "Separable Conjugate Spaces"}, Israel J. Math. \textbf{9} (1971), 279--284. \bibitem{14} J. Lindenstrauss, L. Tzafriri: \textit{Classical Banach spaces, vol.1: Sequence spaces}, Berlin-Heidelberg-New York (1977). \bibitem{15} A. Pietsch: \textit{Eigenwertverteilungen von Operatoren in Banachr\"aumen}, Hausdorff-Festband: Theory of sets and topology, Berlin: Akademie-Verlag (1972), 391-402. \bibitem{16} A. Pietsch: \textit{Operator Ideals}, North Holland (1980). \bibitem{17} A. Pietsch: \textit{Eigenvalues and s-numbers}, Cambridge Univ. Press (1987). \bibitem{18} A. Pietsch: \textit{History of Banach spaces and linear operators}, Birkh\"auser (2007). \bibitem{19} G. Pisier: \textit{Estimations des distances {\`a} un espace euclidien et des constantes de proj{\'e}ction des espaces de Banach de dimensions finie}, Seminaire d'Analyse Fonctionelle 1978-1979, Centre de Math., Ecole Polytech., Paris (1979), expos{\'e} 10, 1--21. \bibitem{20} O.I. Reinov: \textit{A simple proof of two theorems of A. Grothendieck} (Russian), Vestn. Leningr. Univ. \textbf{7} (1983), 115-116. \bibitem{21} O.I. Reinov: \textit{Disappearance of tensor elements in the scale of p-nuclear operators} (Russian), Theory of Operators and Theory of Functions, Leningrad , LGU, vol. 1 (1983), 145-165. \bibitem{22} O.I. Reinov: \textit{A survey of some results in connection with Grothendieck approximation property}, Math. Nachr. \textbf{119} (1984), 257-264. \bibitem{23} O.I. Reinov: \textit{Approximation properties $AP_s$ and p-nuclear operators (the case $0