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 ![]() |
[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 ![]() |
[a8] | M. Talagrand, "La propriété de Dunford–Pettis dans ![]() ![]() |
[a9] | H.P. Lotz, "Tauberian theorems for operators on ![]() |
[a10] | H.P. Lotz, "Uniform convergence of operators on ![]() |
[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 |
Dunford-Pettis property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dunford-Pettis_property&oldid=22367