##
**Absolutely summing operators in \(L_{p}\)-spaces and their applications.**
*(English)*
Zbl 0183.40501

This paper is a milestone of modern functional analysis. It connects the concept of an absolutely \(p\)-summing operator with that of an \({\mathcal L}_p\)-space. Moreover, J. Lindenstrauss and A. Pełczyński made A. Grothendieck’s “Résumé” [Bol.Soc.Mat.São Paulo 8 , 1–79 (1956; Zbl 0074.32303)] accessible to the mathematical community by replacing the language of tensor products by the language of operator ideals (quotation: Though the theory of tensor products has its intrinsic beauty we feel that the results of Grothendieck and their corollaries can be more clearly presented without the use of tensor products). The paper concludes with a long list of stimulating problems.

In what follows, we discuss how the ideas of Lindenstrauss and Pełczyński have influenced the development of Banach space theory during the past 40 years. Let us begin with the most important definition. For \( 1 \leq p \leq \infty\) and \(\lambda \geq1\), a Banach space \(X\) is called an \({\mathcal L}_{p,\lambda}\)-space if every finite-dimensional subspace \(E\) of \(X\) is contained in a finite-dimensional subspace \(F\), \(\dim(F) =n\), whose Banach-Mazur distance to \(l_p^n\) is less than or equal to \(\lambda\). Here \(l_p^n\) denotes the space \(\mathbb{R}^n\) with its canonical \(l_p\)-norm. A Banach space \(X\) is said to be an \({\mathcal L}_p\)-space if it is an \({\mathcal L}_{p,\lambda}\)-space for some \(\lambda \geq1\). According to a theorem of J.T. Joichi [Proc. Am. Math. Soc. 17, 423–426 (1966; Zbl 0193.09001)], the \({\mathcal L}_2\)-spaces are just the Banach spaces isomorphic to a Hilbert space.

\({\mathcal L}_p\)-spaces are those Banach spaces with the same finite-dimensional structure as the Lebesgue spaces \(L_p(M,\mathcal M,\mu)\). This viewpoint of investigating Banach spaces through their finite-dimensional spaces has opened a new stage known as the local theory of Banach spaces that has produced many spectacular results.

The Lindenstrauss-Pełczyński paper contains a fundamental structure theorem which says that every \({\mathcal L}_p\)-space is isomorphic to a complemented subspace of some function space \(L_p(M,\mathcal M,\mu)\). Problem 1c asked whether, conversely, every complemented subspace of \(L_p(M,\mathcal M,\mu)\) is either an \({\mathcal L}_p\)- or an \({\mathcal L}_2\)-space. An affirmative answer was given in a famous paper by J. Lindenstrauss, H. P. Rosenthal [Isr. J. Math. 7, 325–349 (1969; Zbl 0205.12602), Theorem 2.1]. A corollary of this result yields a solution of Problem 1b: A Banach space \(X\) is an \({\mathcal L}_p\)-space if and only if its dual \(X^*\) is an \({\mathcal L}_{p^*}\)-space, \(1/p+1/p^*=1\).

As a by-product of their work about \({\mathcal L}_p\)-spaces, Lindenstrauss and Rosenthal obtained the principle of local reflexivity, which says that a Banach space and its bidual possess, roughly speaking, the same finite-dimensional subspaces. This lemma has become one of the most important tools of Banach space theory.

Unfortunately, some structural conjectures were too optimistic (Problems 1b and 1d). Not every \({\mathcal L}_\infty\)-space is isomorphic to some \(C(K)\). In the case \(1 < p \not= 2 < \infty\), besides \(l_p\), \(l_p \oplus l_2\), \((l_2 \oplus l_2 \oplus \dots)_p\), and \(L_p(0,1)\) there exist infinitely many non-isomorphic separable \({\mathcal L}_p\)-spaces. No classification is known so far. W. B. Johnson, H. P. Rosenthal and M. Zippin [Isr. J. Math. 9 (1971), 488–506 (1971; Zbl 0217.16103), Theorem 5.1] showed that every separable \({\mathcal L}_p\)-space has a basis. For \(1 < p \not= 2 < \infty\) it is conjectured that there even exists an unconditional basis. This is a special case of Problem 7c: Let \(X\) be any Banach with an unconditional basis. Does every complemented subspace of \(X\) have an unconditional basis as well?

The main achievement was the observation that Grothendieck’s théorème fondamental de la théorie métrique des produits tensoriels is equivalent to the following assertion, henceforth known as Grothendieck’s inequality: Let \(\{a_{ij}\}_{i,j=1}^n\) be a finite matrix of real numbers such that \[ \biggl|\sum_{i,j=1}^n a_{ij} s_i t_j \biggr| \leq \sup_i |s_i| \sup_j |t_j| \] for all real numbers \(s_1,\dots, s_n\) and \(t_1,\dots, t_n\). Then \[ \biggl|\sum_{i,j=1}^n a_{ij} \langle x_i,y_j \rangle \biggr| \leq K_G \sup_i \|x_i\| \sup_j \|y_j\| \] for all choices of elements \(x_1,\dots, x_n\) and \(y_1,\dots, y_n\) in a real Hilbert space. The exact value of the best possible constant \(K_G\) (which does not depend on \(n\)) is still unknown.

Another restatement of the fundamental theorem already known to Grothendieck reads as follows: Every operator from an \({\mathcal L}_1\)-space into an \({\mathcal L}_2\)-space is absolutely summing. Problem 2 asked whether the converse implication holds: Let \(X\) and \(Y\) be infinite-dimensional Banach spaces such that every operator \(T\) from \(X\) into \(Y\) is absolutely summing. Does it follow that \(X\) is an \({\mathcal L}_1\)-space and that \(Y\) is an \({\mathcal L}_2\)-space? S. Kislyakov [J. Sov. Math. 16, 1181–1184 (1981; Zbl 0347.46012, Zbl 0459.46009)] and G. Pisier [Ann. Inst. Fourier 28, 69–90 (1978; Zbl 0363.46019)] proved that the answer concerning \(X\) is negative. Finally, according to G. Pisier [Acta Math. 151, 181–208 (1983; Zbl 0542.46038), Remark 4.6] there exists a non-\({\mathcal L}_p\)-space space \(X\) such that all operators from \(X\) into \(X^*\) are even integral.

To the best of my knowledge, Problem 3 is still open: Let \(X\) and \(Y\) be infinite-dimensional Banach spaces such that every operator \(T\) from \(X\) into \(Y\) is absolutely \(p\)-summing for some fixed \(p\), \(1 < p < 2\). Does it follows that every operator from \(X\) into \(Y\) is absolutely summing?

Using Grothendieck’s theorem, Lindenstrauss and Pełczyński proved that every operator from an \({\mathcal L}_\infty\)-space into an \({\mathcal L}_p\)-space with \(1 \leq p \leq 2\) is absolutely \(2\)-summing. Later, E. Dubinsky, A. Pełczyński, H.P. Rosenthal [Stud.Math.44, 617–648 (1972; Zbl 0262.46018)] studied the class of all Banach spaces \(X\) such that every operator from an \({\mathcal L}_\infty\)-space into \(X\) is absolutely \(2\)-summing. These investigations finally led to the concept of cotype \(2\), a key notion of modern Banach space theory.

Problem 4 asked whether every operator from an \({\mathcal L}_\infty\)-space into an \({\mathcal L}_p\)-space is absolutely \(p\)-summing for \(2 < p < \infty\). S. Kwapień [Stud.Math.38, 193–201, Theorem 7 (1970; Zbl 0211.43505)] observed that this conjecture is “almost” true: though such operators may fail to be absolutely \(p\)-summing, they are absolutely \((p+\varepsilon)\)-summing, \(\varepsilon > 0\).

Problem 5 was suggested by the fact that every absolutely \(p\)-summing operator is weakly compact. So it made sense to ask for which exponents \( 1 \leq q < p <\infty\) all absolutely \((p,q)\)-summing operators are weakly compact as well. A first step in answering this question was the observation that the summation operator \(\sigma: (\xi_k) \mapsto (\sum_{k=1}^n \xi_k)\) is universal among all operators \(T: X \to Y\) that fail to be weakly compact; more precisely, there is a factorisation \(\sigma: l_1 \overset{A}{\to} X \overset{T}{\to} Y \overset{B}{\to} l_\infty\). Thus \(\sigma:l_1 \to l_\infty\) was identified as the favourite candidate for a counterexample; and, indeed, S. Kwapień, A. Pełczyński [Stud.Math.34, 43–68, Prop. 4.2 (1970; Zbl 0189.43505)] were able to show that \(\sigma:l_1 \to l_\infty\) is absolutely \((p,q)\)-summing whenever \( 1 \leq q < p <\infty\).

An operator is called Hilbertian if it admits a factorization \(T:X \overset{A}{\to} H \overset{B}{\to} Y\), where \(H\) is a Hilbert space. The class of Hilbertian operators is an injective, surjective, and symmetric ideal. Theorem 5.2 says that for \(1 \leq q \leq 2 \leq p \leq \infty\), every operator from an \({\mathcal L}_p\)-space into an \({\mathcal L}_q\)-space is Hilbertian. Problem 6 is concerned with the following generalization: Let \( 1 \leq q \leq r \leq p \leq\infty\). Does every operator from an \({\mathcal L}_p\)-space into an \({\mathcal L}_q\)-space factor through an \({\mathcal L}_r\)-space? An affirmative answer was given in a fundamental paper by S. Kwapień [Bull.Soc.Math.France, Suppl., Mém.31–32, 215–225, Theorem 1 (1972; Zbl 0246.47040)].

The paper provides several characterizations of \({\mathcal L}_2\)-spaces. For example, a Banach space is isomorphic to a Hilbert space if and only if it is isomorphic to a subspace of an \({\mathcal L}_1\)-spaces and to a quotient of an \({\mathcal L}_\infty\)-space. This criterion inspired S. Kwapień to the proof of his famous type-cotype theorem [Stud.Math.38, 277–278, Prop. 3.1 (1970; Zbl 0256.46024)].

Local techniques may yield global conclusions. Indeed, applying the theory of absolutely \(p\)-summing operators, Lindenstrauss and Pełczyński proved that all normalized unconditional bases of \(l_1\) and \(c_0\) are equivalent. Subsequently, J. Lindenstrauss, M. Zippin [J. Funct.Anal.3, 115–125 (1969; Zbl 0174.17201)] completed the picture by showing that \(l_2\), \(l_1\), and \(c_0\) are the only spaces with this property.

The theory of \({\mathcal L}_p\)-spaces was developed in a short period after their introduction. A comprehensive presentation can be found in [J. Lindenstrauss, L. Tzafriri, “Classical Banach Spaces.” Lecture Notes in Mathematics 338. Springer (1973; Zbl 0259.46011)]. H.P. Rosenthal, G. Schechtman, J. Bourgain and F. Delbaen constructed further examples; see [J. Bourgain, “New Classes of \(\mathcal{L}_p\)-spaces”. Lecture Notes in Mathematics 889. Springer (1981; Zbl 0476.46020)]. But this was not the end of the story, since S. Argyros and R. Haydon [“A hereditarily indecomposable \(\mathcal{L}_\infty\)-space that solves the scalar-plus-compact problem”. Preprint arXiv:0903.3921 (2009)] very recently discovered a separable \({\mathcal L}_\infty\)-space (whose dual is isomorphic to \(l_1\)) in which every operator is of the form \(\lambda I +K\) with \(\lambda \in \mathbb C\) and a compact operator \(K\). This example solves a fascinating problem posed by J. Lindenstrauss almost 40 years ago.

Finally, I quote from my review [MR 37#6743] written in 1969: \(\dots\) die Fülle der Ergebnisse dieser ganz hervorragenden Arbeit konnte durch das vorliegende Referat nur angedeutet werden. The past developments have corrobotated this evaluation again and again.

In what follows, we discuss how the ideas of Lindenstrauss and Pełczyński have influenced the development of Banach space theory during the past 40 years. Let us begin with the most important definition. For \( 1 \leq p \leq \infty\) and \(\lambda \geq1\), a Banach space \(X\) is called an \({\mathcal L}_{p,\lambda}\)-space if every finite-dimensional subspace \(E\) of \(X\) is contained in a finite-dimensional subspace \(F\), \(\dim(F) =n\), whose Banach-Mazur distance to \(l_p^n\) is less than or equal to \(\lambda\). Here \(l_p^n\) denotes the space \(\mathbb{R}^n\) with its canonical \(l_p\)-norm. A Banach space \(X\) is said to be an \({\mathcal L}_p\)-space if it is an \({\mathcal L}_{p,\lambda}\)-space for some \(\lambda \geq1\). According to a theorem of J.T. Joichi [Proc. Am. Math. Soc. 17, 423–426 (1966; Zbl 0193.09001)], the \({\mathcal L}_2\)-spaces are just the Banach spaces isomorphic to a Hilbert space.

\({\mathcal L}_p\)-spaces are those Banach spaces with the same finite-dimensional structure as the Lebesgue spaces \(L_p(M,\mathcal M,\mu)\). This viewpoint of investigating Banach spaces through their finite-dimensional spaces has opened a new stage known as the local theory of Banach spaces that has produced many spectacular results.

The Lindenstrauss-Pełczyński paper contains a fundamental structure theorem which says that every \({\mathcal L}_p\)-space is isomorphic to a complemented subspace of some function space \(L_p(M,\mathcal M,\mu)\). Problem 1c asked whether, conversely, every complemented subspace of \(L_p(M,\mathcal M,\mu)\) is either an \({\mathcal L}_p\)- or an \({\mathcal L}_2\)-space. An affirmative answer was given in a famous paper by J. Lindenstrauss, H. P. Rosenthal [Isr. J. Math. 7, 325–349 (1969; Zbl 0205.12602), Theorem 2.1]. A corollary of this result yields a solution of Problem 1b: A Banach space \(X\) is an \({\mathcal L}_p\)-space if and only if its dual \(X^*\) is an \({\mathcal L}_{p^*}\)-space, \(1/p+1/p^*=1\).

As a by-product of their work about \({\mathcal L}_p\)-spaces, Lindenstrauss and Rosenthal obtained the principle of local reflexivity, which says that a Banach space and its bidual possess, roughly speaking, the same finite-dimensional subspaces. This lemma has become one of the most important tools of Banach space theory.

Unfortunately, some structural conjectures were too optimistic (Problems 1b and 1d). Not every \({\mathcal L}_\infty\)-space is isomorphic to some \(C(K)\). In the case \(1 < p \not= 2 < \infty\), besides \(l_p\), \(l_p \oplus l_2\), \((l_2 \oplus l_2 \oplus \dots)_p\), and \(L_p(0,1)\) there exist infinitely many non-isomorphic separable \({\mathcal L}_p\)-spaces. No classification is known so far. W. B. Johnson, H. P. Rosenthal and M. Zippin [Isr. J. Math. 9 (1971), 488–506 (1971; Zbl 0217.16103), Theorem 5.1] showed that every separable \({\mathcal L}_p\)-space has a basis. For \(1 < p \not= 2 < \infty\) it is conjectured that there even exists an unconditional basis. This is a special case of Problem 7c: Let \(X\) be any Banach with an unconditional basis. Does every complemented subspace of \(X\) have an unconditional basis as well?

The main achievement was the observation that Grothendieck’s théorème fondamental de la théorie métrique des produits tensoriels is equivalent to the following assertion, henceforth known as Grothendieck’s inequality: Let \(\{a_{ij}\}_{i,j=1}^n\) be a finite matrix of real numbers such that \[ \biggl|\sum_{i,j=1}^n a_{ij} s_i t_j \biggr| \leq \sup_i |s_i| \sup_j |t_j| \] for all real numbers \(s_1,\dots, s_n\) and \(t_1,\dots, t_n\). Then \[ \biggl|\sum_{i,j=1}^n a_{ij} \langle x_i,y_j \rangle \biggr| \leq K_G \sup_i \|x_i\| \sup_j \|y_j\| \] for all choices of elements \(x_1,\dots, x_n\) and \(y_1,\dots, y_n\) in a real Hilbert space. The exact value of the best possible constant \(K_G\) (which does not depend on \(n\)) is still unknown.

Another restatement of the fundamental theorem already known to Grothendieck reads as follows: Every operator from an \({\mathcal L}_1\)-space into an \({\mathcal L}_2\)-space is absolutely summing. Problem 2 asked whether the converse implication holds: Let \(X\) and \(Y\) be infinite-dimensional Banach spaces such that every operator \(T\) from \(X\) into \(Y\) is absolutely summing. Does it follow that \(X\) is an \({\mathcal L}_1\)-space and that \(Y\) is an \({\mathcal L}_2\)-space? S. Kislyakov [J. Sov. Math. 16, 1181–1184 (1981; Zbl 0347.46012, Zbl 0459.46009)] and G. Pisier [Ann. Inst. Fourier 28, 69–90 (1978; Zbl 0363.46019)] proved that the answer concerning \(X\) is negative. Finally, according to G. Pisier [Acta Math. 151, 181–208 (1983; Zbl 0542.46038), Remark 4.6] there exists a non-\({\mathcal L}_p\)-space space \(X\) such that all operators from \(X\) into \(X^*\) are even integral.

To the best of my knowledge, Problem 3 is still open: Let \(X\) and \(Y\) be infinite-dimensional Banach spaces such that every operator \(T\) from \(X\) into \(Y\) is absolutely \(p\)-summing for some fixed \(p\), \(1 < p < 2\). Does it follows that every operator from \(X\) into \(Y\) is absolutely summing?

Using Grothendieck’s theorem, Lindenstrauss and Pełczyński proved that every operator from an \({\mathcal L}_\infty\)-space into an \({\mathcal L}_p\)-space with \(1 \leq p \leq 2\) is absolutely \(2\)-summing. Later, E. Dubinsky, A. Pełczyński, H.P. Rosenthal [Stud.Math.44, 617–648 (1972; Zbl 0262.46018)] studied the class of all Banach spaces \(X\) such that every operator from an \({\mathcal L}_\infty\)-space into \(X\) is absolutely \(2\)-summing. These investigations finally led to the concept of cotype \(2\), a key notion of modern Banach space theory.

Problem 4 asked whether every operator from an \({\mathcal L}_\infty\)-space into an \({\mathcal L}_p\)-space is absolutely \(p\)-summing for \(2 < p < \infty\). S. Kwapień [Stud.Math.38, 193–201, Theorem 7 (1970; Zbl 0211.43505)] observed that this conjecture is “almost” true: though such operators may fail to be absolutely \(p\)-summing, they are absolutely \((p+\varepsilon)\)-summing, \(\varepsilon > 0\).

Problem 5 was suggested by the fact that every absolutely \(p\)-summing operator is weakly compact. So it made sense to ask for which exponents \( 1 \leq q < p <\infty\) all absolutely \((p,q)\)-summing operators are weakly compact as well. A first step in answering this question was the observation that the summation operator \(\sigma: (\xi_k) \mapsto (\sum_{k=1}^n \xi_k)\) is universal among all operators \(T: X \to Y\) that fail to be weakly compact; more precisely, there is a factorisation \(\sigma: l_1 \overset{A}{\to} X \overset{T}{\to} Y \overset{B}{\to} l_\infty\). Thus \(\sigma:l_1 \to l_\infty\) was identified as the favourite candidate for a counterexample; and, indeed, S. Kwapień, A. Pełczyński [Stud.Math.34, 43–68, Prop. 4.2 (1970; Zbl 0189.43505)] were able to show that \(\sigma:l_1 \to l_\infty\) is absolutely \((p,q)\)-summing whenever \( 1 \leq q < p <\infty\).

An operator is called Hilbertian if it admits a factorization \(T:X \overset{A}{\to} H \overset{B}{\to} Y\), where \(H\) is a Hilbert space. The class of Hilbertian operators is an injective, surjective, and symmetric ideal. Theorem 5.2 says that for \(1 \leq q \leq 2 \leq p \leq \infty\), every operator from an \({\mathcal L}_p\)-space into an \({\mathcal L}_q\)-space is Hilbertian. Problem 6 is concerned with the following generalization: Let \( 1 \leq q \leq r \leq p \leq\infty\). Does every operator from an \({\mathcal L}_p\)-space into an \({\mathcal L}_q\)-space factor through an \({\mathcal L}_r\)-space? An affirmative answer was given in a fundamental paper by S. Kwapień [Bull.Soc.Math.France, Suppl., Mém.31–32, 215–225, Theorem 1 (1972; Zbl 0246.47040)].

The paper provides several characterizations of \({\mathcal L}_2\)-spaces. For example, a Banach space is isomorphic to a Hilbert space if and only if it is isomorphic to a subspace of an \({\mathcal L}_1\)-spaces and to a quotient of an \({\mathcal L}_\infty\)-space. This criterion inspired S. Kwapień to the proof of his famous type-cotype theorem [Stud.Math.38, 277–278, Prop. 3.1 (1970; Zbl 0256.46024)].

Local techniques may yield global conclusions. Indeed, applying the theory of absolutely \(p\)-summing operators, Lindenstrauss and Pełczyński proved that all normalized unconditional bases of \(l_1\) and \(c_0\) are equivalent. Subsequently, J. Lindenstrauss, M. Zippin [J. Funct.Anal.3, 115–125 (1969; Zbl 0174.17201)] completed the picture by showing that \(l_2\), \(l_1\), and \(c_0\) are the only spaces with this property.

The theory of \({\mathcal L}_p\)-spaces was developed in a short period after their introduction. A comprehensive presentation can be found in [J. Lindenstrauss, L. Tzafriri, “Classical Banach Spaces.” Lecture Notes in Mathematics 338. Springer (1973; Zbl 0259.46011)]. H.P. Rosenthal, G. Schechtman, J. Bourgain and F. Delbaen constructed further examples; see [J. Bourgain, “New Classes of \(\mathcal{L}_p\)-spaces”. Lecture Notes in Mathematics 889. Springer (1981; Zbl 0476.46020)]. But this was not the end of the story, since S. Argyros and R. Haydon [“A hereditarily indecomposable \(\mathcal{L}_\infty\)-space that solves the scalar-plus-compact problem”. Preprint arXiv:0903.3921 (2009)] very recently discovered a separable \({\mathcal L}_\infty\)-space (whose dual is isomorphic to \(l_1\)) in which every operator is of the form \(\lambda I +K\) with \(\lambda \in \mathbb C\) and a compact operator \(K\). This example solves a fascinating problem posed by J. Lindenstrauss almost 40 years ago.

Finally, I quote from my review [MR 37#6743] written in 1969: \(\dots\) die Fülle der Ergebnisse dieser ganz hervorragenden Arbeit konnte durch das vorliegende Referat nur angedeutet werden. The past developments have corrobotated this evaluation again and again.

Reviewer: Albrecht Pietsch (Jena)

### MSC:

46Bxx | Normed linear spaces and Banach spaces; Banach lattices |

46Gxx | Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces) |

47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |