---
author:
- 'Oleg I. Reinov${{ }^\dag}$'
title: 'On factorization of operators through the spaces $ \lp$ '
---

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Oleg I. Reinov Department of MathematicsSt Petersburg UniversitySt Peterhof, Bibliotech pl 2198904 St Petersburg, Russia

oreinorein.usr.pu.ru

${{ }^\dag}$This work was done with partial support of the Ministry of the general and professional education of Russia (Grant 97-0-1.7-36) and FCP "Integracija", reg. No. 326.53.

We give conditions on a pair of Banach spaces $ X$ and $ Y,$ under which each operator from $ X$ to $ Y,$ whose second adjoint factors compactly through the space $ l^p,\,$ $ 1\le p\le+\infty,\,$ itself compactly factors through $ l^p.$ The conditions are as follows: either the space $ X^*,$ or the space $ Y^{***}$ possesses the Grothendieck approximation property. Leaving the corresponding question for parameters $ p>1,\,p\neq 2,$ still open, we show that for $ p=1$ the conditions are essential.

""<span>${ }^\ddag$ AMS Subject Classification: 47B10. Hilbert–Schmidt operators, trace class operators, nuclear operators, p-summing operators, etc.</span>

""<span>${ }$ Key words: $p$-compact operators, bases, approximation property, tensor products.</span>

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One of few questions, which remain open in the theory of operator ideals and are connected with the notion of their regularity, is the question of the possibility of a factorization of a continuous operator, acting between Banach spaces, through the classical sequence spaces $ l^p$ under the condition that this operator, considered as an operator with values in the second dual to the space of images, admits a factorization of such a kind. More precisely, the question is connected with the possibility of the "compact factorization" through these spaces (and we will deal with this "compact" situation). Clearly, we reject the trivial case where $ p=2,$ since in this case the answer is affirmative by evident reasons: every (closed) subspace of a Hilbert space is complemented.

A few of known facts concerning the formulated question is two results obtained by the author yet in 1982 (see \[1\], Corollaries 3.4 and 4.2): if $ p=1$ or $ p=+\infty$ (the factorization through the spaces $ l^1$ and $ c_0)$ then the answer is negative. The technique developed in \[1\] allowed us to construct the corresponding counterexamples in the spaces which, in general, do not possess the Grothendieck approximation property.

In this note we shall give sufficient (and likely close to necessary) conditions, imposed on Banach spaces under consideration, under which the answer to the mentioned question is positive for each value of the parameter $p,\, 1\le p\le \infty.$ On the other hand we will show that the conditions which appear bellow can not be rejected (or weakened). But, unfortunately, the counterexample that we will give concerns only one case, — the case of the compact factorization through the space $ l^1.$

As before, <span>*the question on the $l^p$-factorization of operators in the case where $ p>1,\, p\neq 2,$ still*</span> (for a long time) <span>*remains open*</span>.

It seems to me that I have a counterexample, similar to that of Theorem 2 bellow, for the case of the factorization through the space $ c_0,$ but since the corresponding proof is not completely checked up yet, we do not consider it in the note.

We hold standard notation of the geometrical theory of operators in Banach spaces. The classical reference book on the theory of operator ideals is the A. Pietsch's monograph \[2\]. All the spaces under consideration are Banach spaces. We usually denote the elements of the spaces by the corresponding small letters: $x\in X,\,y\in Y,\dots,\, x'\in X^*,\,y''\in Y^{**},\dots$. For $ p\in [1,+\infty],$ the conjugate exponent $ p'$ is defined by the relation $ 1/p+1/p'=1.$ By $\L\left(X,Y \right)$ it is denoted the space of all (linear continuous) operators from $X$ to $Y$ with its standard norm (generally, in this paper the term "operator" always means "linear continuous operator").

Recall that <span>*a Banach space has the approximation property*</span> (of A. Grothendieck) if the identity map on the space can be approximated, in the topology of compact convergence, by finite dimensional operators (some details and other reformulations of this definition can be found either in the book \[2\] or directly in the origin \[3\]). And one more important reminder. If $ J$ is an operator ideal, then $ J^{ \operatorname{ reg}}(X, Y)$ denotes the space of all operators $ T$ from $ X$ to $ Y,$ for which $ \pi_Y\,T\in J(X, Y^{**}),$ where $ \pi_Y$ is the canonical isometric imbedding of the space $ Y$ into its second dual $ Y^{**}.$ In some cases it will be convenient to consider some of the Banach spaces as the subspaces of their second duals (under the canonical imbeddings), without mentioning the operators $ \pi$ which give these imbeddings.

Let $ 0<p\le\infty.$ A family $ \left\{ x_j\right\}_{j=1}^\infty,$ where $ x_j\in X,$ is said to be <span>*weakly $ p$-summable*</span> if $ \left\{ \<x_j, x' \>\right\}\in l^p$ for all $ x'\in X^*.$ Following A. Pietsch and his notations, we set $$\bw_p(x_j):=
  \sup_{\|x'\|\le 1} \left(\sum_j \, |\<x_j,\,x'\>|^p\right)^{1/p}.$$

If $ 1\le p\le\infty$ and $ S$ is a finite dimensional operator from $ X$ to $ Y,$ then its $ \K_p^0$-norm is the quantity $\bK_p^0(S):= \inf \left\{ \sup_i |\a_i|\,\bw_p(x'_j)\,\bw_{p'}(y_j)
\right\},$ where the infimum is taken over all <span>*finite*</span> representations of the operator $ S$ in the form $$S=\sum_{j=1}^n \a_j\,x'_j\ot y_j.$$ We shall denote by $ X^*\tto_p Y$ the completion of the algebraic tensor product $ X^*\ot Y$ (which we consider as a linear space of all finite dimensional operators) in this norm $ \bK_p^0,$ keeping the same notation $ \bK_p^0$ for the norm in this completion. Each tensor element $ z\in  X^*\tto_p Y$ can be represented in the form of a converging series $ z=\sum_{j=1}^\infty \a_j\,x'_j\ot y_j,$ where $ \a_j\to 0,$ and, moreover, for a given $ \e>0$ this representation can be chosen in such a way that $$\sup_i |a_i|\,\bw_p(x'_j)\,\bw_{p'}(y_j) \le \bK_p^0(z)+\e.$$

The tensor product $ X^*\tto_p Y$ generates naturally a linear subspace of operators in $ \L(X,Y);$ the operator corresponding to an element $ z$ will be denoted by $ \tilde z.$ It is quite possible (but, in general, it is not known whether it is), that some of <span>*different tensors*</span> $ z_1, z_2$ may be "pasting together" when turning into the operators (i. e. for $ z_1\neq z_2$ it is not forbidden for the equality $ \tilde z_1 = \tilde z_2$ to be valid). We denote (again, following the notation of A. Pietsch) by $ \K_p(X,Y)\,$ the factor space of the considered tensor product over the kernel (if non trivial) of the mentioned mapping and by $ \bK_p$ the obtained norm on this space of operators. Note that $ \[\K_p, \bK_p\] $ is a normed operator ideal and it is a particular case of the so called ideals of $ (r, p ,q)$-nuclear operators, — $ \[\N_{(r,p,q)}, \bN_{(r,p,q)}\]\!:$  $ \[\K_p, \bK_p\] =  \[\N_{(\infty,p,p')}, \bN_{(\infty,p,p')}\].$ For details we refer the reader to the book \[2\]. However, there is a thing that is very important for us and which is concerned with the possibility of the mentioned nontrivial factorization. Namely, <span>*if one of the space $ X^*$ or $ Y$ has the approximation property then the following <span>(formal)</span> equality holds*</span>: $ \[X^*\tto_p Y, \bK_p^0\] = \[\K_p(X,Y), \bK_p\],$ that is the kernel of the canonical map $ X^*\tto_p Y \to \K_p(X, Y)$ is trivial and this map is an isometric bijection. The proof of this fact is now standard, so we omit it suggesting to the reader as a not difficult exercise.

We say that an operator $ T: X\to Y$ <span>*compactly factors through the space $ l^p$*</span> (<span>*$p$-compact*</span> in the terminology of A. Pietsch), if there exist two compact operators $ A: X\to l^p$ and $ B: l^p\to Y$ for which $ T=BA.$ As a norm of such an "$l^p$-factored operator" we take the quantity $ \inf\,\|A\|\,\|B\|,$ where the infimum is taken over all possible factorizations of the operator $ T$ of the mentioned kind.

One of the central theorem in the theory of compactly $ l^p$-factored operators is formulated with the help of the following relations: $$\bK_p(T) = \inf\,\{\|A\|\,\|B\|:\, T=BA: X\to l^p\to Y;\, A,B \,
\text{ are compact} \}$$ and <span>*the ideal $ \K_p$ coincides with the ideal of all operators which can be compactly factored through* </span> $ l^p$ (see \[2\], Theorem 18.3.2). The space $ \D_{p'}(Y,X^{**})\, $ which is conjugate to $ X^*\tto_p Y$ is described in the same place of the mentioned monograph, but since, as a matter of fact, we need not to know of what operators this conjugate space consists, we are not going to give more details. It is worthwhile, however, for the reader to look in the book for getting a minimal information about the elements of the conjugate space. For us it is important to know that from the definition of the operators lying in this space it follows immediately that <span>*all the series that will appear in the proof of our first theorem are convergent*</span> (so we won't have to be distracted every time to check up the convergence of each appearing series).

Let $\, p\in [1,+\infty].$ If one of the space $\, X^* $ or $\, Y^{***} $ has the Grothendieck approximation property, then any operator from $ X$ into $ Y,$ whose second adjoint factors compactly through the space $ l^p,$ itself compactly factors through $ l^p.$ Thus, in this case $ \K^{ \operatorname{ reg}}_p (X, Y)= \K_p (X, Y).$

Suppose that there exists such an operator $ T\in \L(X,Y)$ that $ T\notin \K_p(X,Y),$ but $ \pi_Y\,T\in \K_p(X,Y^{**}).$ Since either $ X^*$ or $ Y^{**}$ has the approximation property, then $ \K_p(X,Y^{**})=X^*\tto_p Y^{**}.$ Therefore, the operator $ \pi_Y\,T$ can be identified with a tensor element $ t\in X^*\tto_p Y^{**};$ in addition, by the choice of $ T,$ $  t\notin X^*\tto_p Y$  (here $  X^*\tto_p Y$ is considered as a subspace of the space $  X^*\tto_p Y^{**}$). Hence, there exists such an operator $ U\in \D_{p'}(Y^{**},X^{**})=\( X^*\tto_p Y^{**}\)^*$ that $ \tr U\circ t=\tr \(t^*\circ \( U^*|_{X^*}\) \)=1$ and $ \tr U\circ \pi_Y\circ z=0$ for every $ z\in X^*\tto_p Y.$ &gt;From the last it follows, in particular, that $$U|_{\pi_Y(Y)}=0 \, \text{ É }\,  \pi_Y^*\,U^*|_{X^*}=0. \tag1$$ Indeed, if $ x'\in X^*$ and $ y\in Y,$ then $$\<U\pi_Y\,y,x'\> = \<y, \pi_Y^*\,U^*|_{X^*}x'\> =
\tr \,(U\pi_Y)\circ (x'\otimes y)=0.$$ Evidently, the tensor element $ U\circ t$ generates the operator $ U\pi_Y T$ which is, by the previous reasoning, equal identically to zero.

If the space $ X^*$ has the approximation property then $ X^*\tto_p X^{**}= \K_p(X,X^{**})$ and so the tensor element $ U\circ t$ is zero, in contradiction with the equality $ \tr\, U\circ t=1.$

Let now $ Y^{***}$ has the approximation property. In this case $ V:= \( U^*|_{X^*}\)\circ T^*\circ \pi_Y^*:$ $$\CD    Y^{***} @>\pi_Y^* >> Y^* @> T^* >> X^* @> U^*|_{X^*} >> Y^{***}
\endCD$$ uniquely determines a tensor element $ t_0\in Y^{****}\tto_{p'} Y^{***}.$ Let us take any representation $ t=\sum \a_n x'_n\otimes y''_n$ for $ t$ as an element of the space $ X^*\tto_p Y^{**}.$ Denoting for the simplicity by $ U_*$ the operator $ U^*|_{X^*}$ and recalling that $ \pi_Y T=t,$ we get: $$\multline
   Vy'''=U_*\, \( T^*\pi_Y^*\,y'''\) =
    U_*\, \( (T^*\pi_Y^*\pi_{Y^*})\,\pi_Y^*\, y'''\) = \\ =
    U_*\, \( (\pi_Y T)^*\,\pi_{Y^*})\,\pi_Y^*\, y'''\) =
    U_*\, \( \(\sum \a_n y''_n\otimes x'_n\)\,\pi_{Y^*}\,\pi_Y^*\, y'''\)
  = \\ =U_*\, \( \sum \a_n \<y''_n, \pi_Y^*\, y'''\> \,x'_n \) =
      \sum \a_n \<\pi_Y^{**}y''_n,  y'''\> \,U_* x'_n.
  \endmultline$$

Thus, the operator $ V$ (or the element $ t_0$) has in the space $ Y^{****}\tto_{p'} Y^{***}$ the representation $$V= \sum \a_n \pi_Y^{**}(y''_n)\otimes U_* (x'_n).$$ Therefore (see (1)), $$\tr t_0=\tr V= \sum \a_n \<\pi_Y^{**}(y''_n), U_* (x'_n)\> =
        \sum \a_n \<y''_n, \pi_Y^*\,U_* x'_n\> =  \sum 0=0.$$ On the other hand, $$\multline
 Vy'''= U_* \( \pi_Y T\)^* y'''= U_*\circ t^* (y''')=\\
     =U_*\, \( \sum \a_n \<y''_n, y'''\> \, x'_n\)=
      \sum \a_n \<y''_n, y'''\> \, U_* x'_n,
\endmultline$$ whence $ V=\sum y''_n\otimes U_*(x'_n).$ Therefore $$\tr t_0=\tr V= \sum \a_n \<y''_n, U_* x'_n\> =
 \sum \a_n \<Uy''_n, x'_n\> = \tr U\circ t=1.$$ The obtained contradiction completes the proof of the theorem. $\quad\blacksquare$

Our next result shows that, in general, the approximation conditions imposed on $ X$ and $Y\,$ in Theorem 1 are essential and can not be replaced by weaker conditions "one of the space $ X$ or $ Y^{**}$ has the (bounded) approximation property".

There exists a Banach space $Z,\,$ with the following properties: i<span>)</span> all the duals spaces of $ Z$ are separable; ii<span>)</span> $Z^{**}\,$ has a boundedly complete basis; iii<span>)</span> $Z^{***}\,$ does not have the approximation property; iv<span>)</span> there exists an operator $ T\in\L(Z^{**},Z)$ that is not factored compactly through the space $ l^1,$ and for which, however, the operator $ \pi_Z\,T: Z^{**}\to Z\hookrightarrow Z^{**}$ compactly factors through $ l^1.$ Thus, $ \K_1(Z^{**}, Z)\subsetneqq  \K_1^{ \operatorname{ reg}}(Z^{**}, Z).$ We use the following result obtained in the paper \[1\] (see Lemma 1.2,2) and Corollary 1.2) in \[1\]): there exists a reflexive separable Banach space $ E$ such that for every $ r,\,$ $ 1\le r\le \infty\,$ $ r\neq 2,$ the canonical map $ E^*\wh\ot_r E\to \N_r(E, E)$ is not one-to-one. Here $ E^*\wh\ot_r E$ is the $ r$-projective tensor product associated with the space $ \N_r(E, E)$ of all $ r$-nuclear operators in $ E.$ We consider the case $ r=\infty$ and, passing to the conjugate spaces, formulate the dual result (setting $ X:=E^*$): there exists a reflexive separable Banach space $ X$ such that the canonical mapping $ X^*\tto_1 X\to \K_1(X, X)$ is not one-to-one.

By using the theorem 1 and the corollary 1 from the paper \[4\] (see also the proof of the mentioned corollary), find such a separable Banach space $ Z$ that $ Z^{**}$ has a boundedly complete basis, the space $ X$ is isomorphic to the factor space $ Z^{**}/\pi_Z(Z)$ and, in addition, the corresponding linear homomorphism $ \ffi: Z^{**}\to X$ (with the kernel $ \operatorname{ Ker \ffi}= \pi_Z(Z)$) has the property that the subspace $ \ffi^*(X^*)$ is complemented in $ Z^{***}.$

Let $ u\in  X^*\tto_1 X$ be a nonzero tensor element generating an operator $ \wt u$ which is identically zero. Let us consider any representation of the tensor $ u$ in the form $$u= \sum_{j=1}^\infty \a_j\,x'_j\ot x_j,$$ where the sequence $ ( x_j)$ is bounded, the sequence $ ( x'_j)$ is weakly 1-summable and $ \a_j\to 0.$ We can lift up the elements $ x_j$ to the elements $ z''_j$ of the space $ Z^{**}$ in such a way that for every $ j\in\Bbb N$ $ \ffi(z''_j)=x_j$ and the sequence $ (z''_j)$ is bounded in $ Z^{**}.$ Arising from $ u$ a new tensor element $$v = \sum_{j=1}^\infty \a_j\,x'_j\ot z''_j,$$ belongs to the tensor product $ X^*\tto_1 Z^{**}$ and is not equal to zero, and also the composition $ u=\ffi\circ v,$ considered as an operator, is zero. Since the space $ Z^{**}$ has a basis (and, in particular, has the approximation property), the element $ v$ can be correctly identified with the associated operator $ \wt v: X\to Z^{**},$ which is thus not zero and for which, as said above, the image $ \wt v(X)$ lies in the subspace $ \pi_Z(Z),$ the kernel of the operator $ \ffi.$ Thus, we get an operator $ V: X\to Z$ for which $V=\pi^{-1}_Z \wt v_0 $ and $ V^{**}=\wt v,$ where $ \wt v_0: X\to \pi_Z(Z)$ is the operator induced by $ \wt v.$ It is not difficult to see that the operator $ V$ can not belong to the space $ \K_1(X, Z)=X^*\tto_1 Z,$ since else, for each its (nonzero!) tensor representation in this space, its image under acting of the homomorphism $ \ffi$ would be a zero tensor in $ X^*\tto_1 X$ (we identify now the spaces $ Z$ and $ \pi_Z(Z)$); and on the other hand this image is just $ u.$ So, we can conclude that $X^*\tto_1 Z = \K_1(X, Z)\subsetneqq \K_1^{ \operatorname{ reg}}(X, Z). $

Finally, recalling that the subspace $ \ffi^*(X^*)$ is complemented in $ Z^{***},$ we see that $ \K_1(Z^{**}, Z) =
Z^{***}\tto_1 Z\subsetneqq \K_1^{ \operatorname{ reg}}(Z^{**}, Z).$ $\quad\blacksquare$

REFERENCES

\[\]1 O. I. Reinov 125-134 Approximation properties of order p and the existence of non-p-nuclear operators with p-nuclear second adjoints 1982 109 Math. Nachr.

\[\]2 A. Pietsch Operator ideals North-Holland1980

\[\]3 A. Grothendieck 196+140 Produits tensoriels topologiques et espases nucléaires 195516 Mem. Amer. Math. Soc.

\[\]4 L. Lindenstrauss 279-284 On James' paper "Separable Conjugate Spaces" 19719 Israel J. Math.

{'source': 'arXiv_out/math0107153'}