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On the Dunford-Pettis property of the tensor product of C(K) spaces
Abstract
In this paper we characterize those compact Hausdorff spaces such that (and ) have the Dunford-Pettis Property, answering thus in the negative a question posed by Castillo and González who asked if and have this property.
... Several authors studied whether the DP P of E and F implies the DP P of their projective tensor product E⊗ π F and of their injective tensor product E⊗ F ([1], [5], [9], [12], [13], [21], [23], [26], [28], [36], and [41]). Talagrand [41] gave an example of a Banach space E so that E * has the Schur property, but C([0, 1], E) C[0, 1] ⊗ E and L 1 ([0, 1], E * ) L 1 [0, 1] ⊗ π E * do not have the DP P . ...
... Ryan [36] proved that if E and F have the DP P and contain no copy of 1 , then E ⊗ π F has the DP P and contains no copy of 1 . Bombal and Villanueva proved that if K 1 and K 2 are infinite compact Hausdorff spaces, then C(K 1 ) ⊗ π C(K 2 ) has the DP P if and only if both K 1 and K 2 are scattered ( [5], Theorem 2.2). González and Gutiérrez proved that if E does not have the Schur property, F contains a copy of 1 , and L(E, F * ) = CC(E, F * ), then E ⊗ π F does not have the DP P ( [28], Theorem 3). ...
... We use the sequential characterization of the DP P [19] to show that in some cases, these spaces fail to have the DP P . Results in this paper generalize results in [5] and [28]. ...
We give sufficient conditions on Banach spaces E and F so that their projective tensor product E⊗ πF and the duals of their projective and injective tensor products do not have the Dunford-Pettis property. We prove that if E* does not have the Schur property, F is infinite-dimensional, and every operator T:E*→F** is completely continuous, then (E⊗ {small element of}F)* does not have the DPP. We also prove that if E* does not have the Schur property, F is infinite-dimensional, and every operator T:F**→E* is completely continuous, then (E⊗ πF)*≃L(E,F*) does not have the DPP.
... As a counter-demonstration, Corollary 2 above can easily be applied to establish the existence of non-completely continuous operators in many cases. For example, Theorem 2.2 of [8] shows that if K 1 and K 2 are infinite compact Hausdorff spaces, then C(K 1 ) ⊗ π C(K 2 ) has the DP P if and only if both K 1 and K 2 are scattered ( 1 → C(K 1 ) and 1 → C(K 2 )). Therefore, if either K 1 or K 2 is not scattered, Corollary 2 immediately shows that L(C(K 1 )⊗ π C(K 2 ), c 0 ) = CC(C(K 1 )⊗ π C(K 2 ), c 0 ). ...
... Another application of Corollary 2 shows that in this case L(E ⊗ π F, c 0 ) = CC(E ⊗ π F, c 0 ). Lemma 12. ( [8]) Suppose that L(E, F * ) = CC(E, F * ), (x n ) is a weakly null sequence in E and (y n ) is a bounded sequence in F . Then the sequence (x n ⊗ y n ) is weakly null in E ⊗ π F . ...
... We observe that L( ∞ , c 0 ) = CC( ∞ , c 0 ) (by Corollary 5 (i)), but L( ∞ ⊗ π ∞ , c 0 ) = CC( ∞ ⊗ π ∞ , c 0 ) (by Corollary 14). Moreover, ∞ ⊗ π ∞ does not have the DP P [8]. ...
A Banach space X has the Dunford-Pettis property (DPP) provided that every weakly compact operator T from X to any Banach space Y is completely continuous (or a Dunford-Pettis operator). It is known that X has the DPP if and only if every weakly null sequence in X is a Dunford-Pettis subset of X. In this paper we give equivalent characterizations of Banach spaces X such that every weakly Cauchy sequence in X is a limited subset of X. We prove that every operator T : X -> c(0) is completely continuous if and only if every bounded weakly precompact subset of X is a limited set. We show that in some cases, the projective and the injective tensor products of two spaces contain weakly precompact sets which are not limited. As a consequence, we deduce that for any infinite compact Hausdorff spaces K-1 and K-2, C(K-1) 0 circle times(pi) C(K-2) and C(K-1) circle times(epsilon) C(K-2) contain weakly precompact sets which are not limited.
... Moreover, in [11] J. Bourgain provided interesting sufficient conditions for subspaces L of the Banach space C(K) to have the Dunford-Pettis property. These results have been used by J.A. Cima and R.M. Timoney [12] to study the Dunford-Pettis property for T -invariant algebras on K. F. Bombal and I. Villanueva characterized in [6] those compact spaces K such that C(K)⊗C(K) has the Dunford-Pettis property. ...
... Once again applying the Eberlein-Šmulian theorem and passing to a subsequence if needed, we can assume that {µ n } n∈N weakly converges to a measure µ ∈ L ⊆ M c ( [0, κ) ) β . Now (6) and the (sDP ) property of C k ( [0, κ) ) proved above imply ...
For a Tychonoff space X, let Ck(X) and Cp(X) be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that Ck(X) has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that Ck(X) has both the Dunford–Pettis property and the sequential Dunford–Pettis property if X satisfies one of the following conditions: (1) X is a hemicompact space, (2) X is a cosmic space (= a continuous image of a separable metrizable space), (3) X is the ordinal space [0,κ) for some ordinal κ, or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then Ck(X) has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that Cp(X) has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X, and Cp(X) has the Grothendieck property if and only if every functionally bounded subset of X is finite.
... (iii) The Schur property for a Banach space has been characterized in terms of the Bohr compactification of its underlying group in [16]. The Dunford-Pettis property for the vector-valued versions of c 0 , l 1 , C(M ) is investigated, for instance, in [4], [7], [8]. ...
... Moreover, in [11] J. Bourgain provided interesting sufficient conditions for subspaces L of the Banach space C(K) to have the Dunford-Pettis property. This results have been used by J.A. Cima and R.M. Timoney [12] to study the Dunford-Pettis property for T -ivariant algebras on K. F. Bombal and I. Villanueva characterized in [6] those compact spaces K such that C(K)⊗C(K) have the Dunford-Pettis property. ...
For a Tychonoff space X, let and be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A.~Grothendieck states that has the Dunford-Pettis property and the sequential Dunford--Pettis property. We extend Grothendieck's result by showing that has both the Dunford-Pettis property and the sequential Dunford-Pettis property if X satisfies one of the following conditions: (i) X is a hemicompact space, (ii) X is a cosmic space (=a continuous image of a separable metrizable space), (iii) X is the ordinal space for some ordinal , or (vi) X is a locally compact paracompact space. We show that if X is a cosmic space, then has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that has the Dunford--Pettis property and the sequential Dunford-Pettis property for every Tychonoff space X, and has the Grothendieck property if and only if every functionally bounded subset of X is finite.
... The following lemma is known [8]; we include the proof for the convenience of the reader. ...
A Banach space X has the reciprocal Dunford-Pettis property (RDPP) if every completely continuous operator T from X to any Banach space Y is weakly compact. A Banach space X has the RDPP (resp. property (wL)) if every L-subset of X* is relatively weakly compact (resp. weakly precompact). We prove that the projective tensor product X ⊗πY has property (wL) when X has the RDPP, Y has property (wL), and L(X, Y*) = K(X, Y*).
... The bounded sequences (x n +x n ) and (x n ) verify that for every continuous polynomial P on X one has lim P(x n +u n xx n )=0, since X is an M-space. On the other hand, if we construct a 3-linear form on X following the idea of Bombal and Villanueva in [3] as follows : ...
We study which polynomial properties are, and which are not, tree-space properties.
In this paper we study equivalent formulations of the DP∗Pp (1 < p < ∞). We show that X has the DP∗Pp if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give sufficient conditions on Banach spaces X and Y so that the projective tensor product X ⊗π Y, the dual (X ⊗ϵY)∗ of their injective tensor product, and the bidual (X ⊗π Y)∗∗ of their projective tensor product, do not have the DP Pp, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP∗Pp, 1 < p < ∞.
A linear operator between vector spaces is called strictly singular if for any infinite dimensional closed vector subspace M of X, the restriction of T on M is not a topological isomorphism. In this note we introduced some sufficient conditions on domain and range spaces such that any bounded linear operator in between is strictly singular, and give some examples of spaces satisfying these conditions.
We give conditions on a Banach space X so that the spaces of scalar-valued homogeneous polynomials on X do not have the Dunford-Pettis property (DPP). This allows us to obtain new examples of Banach spaces with the DPP such that their duals fail it.
Many fundamental processes in analysis are best understood by studying and comparing the summability of series in various modes of convergence. This text provides the beginning graduate student, one with basic knowledge of real and functional analysis, with an account of p-summing and related operators. The account is panoramic, with detailed expositions of the core results and highly non-trivial applications to, for example, harmonic analysis, probability and measure theory, and operator theory. Graduate students and researchers from real, complex and functional analysis, and probability theory will benefit from this account.
The purpose of this paper is to characterise the class of regular continuous multilinear operators on a product of C(K) spaces, with values in an arbitrary Banach space. This class has been considered recently by several authors in connection with problems of factorisation of polynomials and holomorphic mappings. We also obtain several characterisations of a compact dispersed space K in terms of polynomials and multilinear forms defined on C(K).
Consider the following possible properties which a Banach space X may have: (P): If ( x i ) and ( y j ) are bounded sequences in X such that for all n ≥ 1 and for every continuous n-homogeneous polynomial P on X, P ( x j ) − ( y j ) → 0, then Q ( x j − y j ) → 0 for all m ≥ 1 and for every continuous m-homogeneous polynomial Q on X .
(RP): If ( x j )and ( y j ) are bounded sequences in X such that for all n ≥ 1 and for every continuous n-homogeneous polynomial P on X, P ( x j − y j ) → 0, then Q ( x j ) − Q ( y j ) → 0 for all m ≥ 1 and for every continuous m-homogeneous polynimial Q on X. We study properties ( P ) and ( RP ) and their relation with the Schur proqerty, Dunford-Pettis property, Λ, and others. Several applications of these properties are given.
