Dunford-Pettis property

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

Equivalently, given weakly convergent sequences in and in its topological dual , the sequence also converges. Contrary to intuition this does not always happen. For example, if denotes the canonical basis of , then is weakly convergent to zero although .

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 and any Banach space , every weakly compact operator into 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 which could be proven to have a compact square. The main examples of spaces having the Dunford–Pettis property are the spaces of continuous functions on a compact space and the spaces 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 and its higher duals (cf. also Hardy spaces); the quotient space and its higher duals (the space itself does not have the Dunford–Pettis property, nor does its dual BMO or its pre-dual VMO) (cf. also -space; -space); the ball algebra, the poly-disc algebra and their duals, and the spaces of -smooth functions on the -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 with the Dunford–Pettis property such that and 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 has the Dunford–Pettis property, then so does . From Rosenthal's theorem it follows that if has the Dunford–Pettis property and does not contain , then has the Dunford–Pettis property. Stegall has shown that although the space has the Dunford–Pettis property (since weakly convergent sequences are norm convergent), its dual does not have the Dunford–Pettis property (because it contains complemented copies of ).

A reflexive space does not have the Dunford–Pettis property unless it is finite-dimensional. The Grothendieck spaces , , , and (cf. Grothendieck space) also possess the Dunford–Pettis property (see [a9], [a10]).

A Banach space is a Grothendieck space with the Dunford–Pettis property if and only if every weak- convergent sequence in converges weakly and uniformly on weakly compact subsets of , if and only if every bounded linear operator from into 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 be a Grothendieck space with the Dunford–Pettis property. Then:

1) does not have a Schauder decomposition, or equivalently, if a sequence of projections on converges weakly to the identity operator , then for sufficiently large;

2) if the Cesáro mean of an operator on converges strongly, then it converges uniformly;

3) all -semi-groups on are norm-continuous (see [a9], [a10]);

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

5) for general ergodic systems on , in particular, -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 " 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 " Canad. J. Math. , 5 (1953) pp. 129–173
[a8]	M. Talagrand, "La propriété de Dunford–Pettis dans et " Israel J. Math. , 44 (1983) pp. 317–321
[a9]	H.P. Lotz, "Tauberian theorems for operators on and similar spaces" , Functional Analysis III. Surveys and Recent Results , North-Holland (1984)
[a10]	H.P. Lotz, "Uniform convergence of operators on 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