# Dunford-Pettis property

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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
How to Cite This Entry:
Dunford-Pettis property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dunford-Pettis_property&oldid=53450
This article was adapted from an original article by J.M.F. CastilloS.-Y. Shaw (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article