Dunford-Pettis property

The property of a Banach space $ X $ that every continuous operator $ T : X \rightarrow Y $ sending bounded sets of $ X $ into relatively weakly compact sets of $ Y $( called weakly compact operators) also transforms weakly compact sets of $ X $ into norm-compact sets of $ Y $( such operators are called completely continuous; cf. also Completely-continuous operator). In short, it requires that weakly compact operators on $ X $ are completely continuous.

Equivalently, given weakly convergent sequences $ ( x _ {n} ) $ in $ X $ and $ ( f _ {n} ) $ in its topological dual $ X ^ {*} $, the sequence $ ( f _ {n} ( x _ {n} ) ) _ {n} $ also converges. Contrary to intuition this does not always happen. For example, if $ ( e _ {n} ) $ denotes the canonical basis of $ l _ {2} $, then $ ( e _ {n} ) $ is weakly convergent to zero although $ e _ {n} ( e _ {n} ) = 1 $.

The property was isolated and defined by A. Grothendieck [a7] after the following classical result of N. Dunford and B.J. Pettis [a5]: For any measure $ \mu $ and any Banach space $ Y $, every weakly compact operator $ L _ {1} ( \mu ) $ into $ Y $ is completely continuous.

This result has its roots in examples of Sirvint, S. Kakutani, Y. Mimura and K. Yosida concerning weakly compact non-compact operators on $ L _ {1} ( 0,1 ) $ which could be proven to have a compact square. The main examples of spaces having the Dunford–Pettis property are the spaces $ C ( K ) $ of continuous functions on a compact space and the spaces $ L _ {1} ( \mu ) $ of integrable functions on a measure space, as well as complemented subspaces of these spaces. Other classical function spaces having the Dunford–Pettis property are: the Hardy space $ H ^ \infty $ and its higher duals (cf. also Hardy spaces); the quotient space $ L _ {1} /H ^ {1} $ and its higher duals (the space $ H ^ {1} $ itself does not have the Dunford–Pettis property, nor does its dual BMO or its pre-dual VMO) (cf. also $ { \mathop{\rm BMO} } $- space; $ { \mathop{\rm VMO} } $- space); the ball algebra, the poly-disc algebra and their duals, and the spaces $ C ^ {k} ( T ^ {n} ) $ of $ k $- smooth functions on the $ n $- dimensional torus.

A classical survey on the topic is [a4]. Many of the open problems stated there have been solved by now, mainly by J. Bourgain [a2], [a3], who introduced new techniques for working with the Dunford–Pettis property, and by M. Talagrand [a8], who gave an example of a space $ X $ with the Dunford–Pettis property such that $ C ( K,X ) $ and $ L _ {1} ( \mu,X ^ {*} ) $ fail the Dunford–Pettis property.

The Dunford–Pettis property is not easy to work with, nor is it well understood. In general, it is difficult to prove that a given concrete space has the property; quoting J. Diestel: "I know of no case where the reward (when it comes) is easily attained" . On the question of structure theorems, many open problems remain. One of the most striking is as follows. When does the dual of a space that has the Dunford–Pettis property have the Dunford–Pettis property? It is clear that if $ X ^ {*} $ has the Dunford–Pettis property, then so does $ X $. From Rosenthal's $ l _ {1} $ theorem it follows that if $ X $ has the Dunford–Pettis property and does not contain $ l _ {1} $, then $ X ^ {*} $ has the Dunford–Pettis property. Stegall has shown that although the space $ l _ {1} ( l _ {2} ^ {n} ) $ has the Dunford–Pettis property (since weakly convergent sequences are norm convergent), its dual $ l _ \infty ( l _ {2} ^ {n} ) $ does not have the Dunford–Pettis property (because it contains complemented copies of $ l _ {2} $).

A reflexive space does not have the Dunford–Pettis property unless it is finite-dimensional. The Grothendieck spaces $ C ( \Omega ) $, $ L ^ \infty ( \mu ) $, $ B ( S, \Sigma ) $, and $ H ^ \infty ( D ) $( cf. Grothendieck space) also possess the Dunford–Pettis property (see [a9], [a10]).

A Banach space $ X $ is a Grothendieck space with the Dunford–Pettis property if and only if every weak- $ * $ convergent sequence in $ X ^ {*} $ converges weakly and uniformly on weakly compact subsets of $ X $, if and only if every bounded linear operator from $ X $ into $ c _ {0} $ is weakly compact and maps weakly compact sets into norm-compact sets.

An interesting phenomenon about Grothendieck spaces with the Dunford–Pettis property is that in many cases strong convergence of operators on such a space (cf. also Strong topology) implies uniform convergence. For example, let $ X $ be a Grothendieck space with the Dunford–Pettis property. Then:

1) $ X $ does not have a Schauder decomposition, or equivalently, if a sequence of projections $ \{ P _ {n} \} $ on $ X $ converges weakly to the identity operator $ I $, then $ P _ {n} = I $ for $ n $ sufficiently large;

2) if the Cesáro mean $ n ^ {- 1 } \sum _ {k = 0 } ^ {n - 1 } T ^ {k} $ of an operator $ T $ on $ X $ converges strongly, then it converges uniformly;

3) all $ C _ {0} $- semi-groups on $ X $ are norm-continuous (see [a9], [a10]);

4) all strongly continuous cosine operator functions on $ X $ are norm-continuous [a11];

5) for general ergodic systems on $ X $, in particular, $ C _ {0} $- semi-groups and cosine operator functions, strong ergodicity implies uniform ergodicity (see [a12]).

References

[a1]	J. Bourgain, "On the Dunford–Pettis property" Proc. Amer. Math. Soc. , 81 (1981) pp. 265–272
[a2]	J. Bourgain, "New Banach space properties of the disc algebra and $G^\infty$" Acta Math. , 152 (1984) pp. 1–48
[a3]	J. Bourgain, "The Dunford–Pettis property for the ball-algebras, the polydisc algebra, and the Sobolev spaces" Studia Math. , 77 (1984) pp. 245–253
[a4]	J. Diestel, "A survey or results related to the Dunford–Pettis property" , Contemp. Math. , 2 , Amer. Math. Soc. (1980) pp. 15–60
[a5]	N. Dunford, B.J. Pettis, "Linear operations on summable functions" Trans. Amer. Math. Soc. , 47 (1940) pp. 323–392
[a6]	N. Dunford, J.T. Schwartz, "Linear operators" , I. General theory , Wiley, reprint (1988)
[a7]	A. Grothendieck, "Sur les applications linéaires faiblement compactes d'espaces de type $C(K)$" Canad. J. Math. , 5 (1953) pp. 129–173
[a8]	M. Talagrand, "La propriété de Dunford–Pettis dans $C(K,E)$ et $L_1(E)$" Israel J. Math. , 44 (1983) pp. 317–321
[a9]	H.P. Lotz, "Tauberian theorems for operators on $L^\infty$ and similar spaces" , Functional Analysis III. Surveys and Recent Results , North-Holland (1984)
[a10]	H.P. Lotz, "Uniform convergence of operators on $L^\infty$ and similar spaces" Math. Z. , 190 (1985) pp. 207–220
[a11]	S.-Y. Shaw, "Asymptotic behavior of pseudoresolvents on some Grothendieck spaces" Publ. RIMS Kyoto Univ. , 24 (1988) pp. 277–282
[a12]	S.-Y. Shaw, "Uniform convergence of ergodic limits and approximate solutions" Proc. Amer. Math. Soc. , 114 (1992) pp. 405–411