View source for Dunford-Pettis property

The property of a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d1102401.png" /> that every continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d1102402.png" /> sending bounded sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d1102403.png" /> into relatively weakly compact sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d1102404.png" /> (called weakly compact operators) also transforms weakly compact sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d1102405.png" /> into norm-compact sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d1102406.png" /> (such operators are called completely continuous; cf. also [[Completely-continuous operator|Completely-continuous operator]]). In short, it requires that weakly compact operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d1102407.png" /> are completely continuous.

Equivalently, given weakly convergent sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d1102408.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d1102409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024010.png" /> in its topological dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024011.png" />, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024012.png" /> also converges. Contrary to intuition this does not always happen. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024013.png" /> denotes the canonical basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024015.png" /> is weakly convergent to zero although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024016.png" />.

The property was isolated and defined by A. Grothendieck [[#References|[a7]]] after the following classical result of N. Dunford and B.J. Pettis [[#References|[a5]]]: For any measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024017.png" /> and any Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024018.png" />, every weakly compact operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024019.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024020.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024021.png" /> which could be proven to have a compact square. The main examples of spaces having the Dunford–Pettis property are the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024022.png" /> of continuous functions on a compact space and the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024023.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024024.png" /> and its higher duals (cf. also [[Hardy spaces|Hardy spaces]]); the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024025.png" /> and its higher duals (the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024026.png" /> itself does not have the Dunford–Pettis property, nor does its dual BMO or its pre-dual VMO) (cf. also [[BMO-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024027.png" />-space]]; [[VMO-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024028.png" />-space]]); the ball algebra, the poly-disc algebra and their duals, and the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024030.png" />-smooth functions on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024031.png" />-dimensional torus.

A classical survey on the topic is [[#References|[a4]]]. Many of the open problems stated there have been solved by now, mainly by J. Bourgain [[#References|[a2]]], [[#References|[a3]]], who introduced new techniques for working with the Dunford–Pettis property, and by M. Talagrand [[#References|[a8]]], who gave an example of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024032.png" /> with the Dunford–Pettis property such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024034.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024035.png" /> has the Dunford–Pettis property, then so does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024036.png" />. From Rosenthal's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024038.png" /> theorem it follows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024039.png" /> has the Dunford–Pettis property and does not contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024041.png" /> has the Dunford–Pettis property. Stegall has shown that although the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024042.png" /> has the Dunford–Pettis property (since weakly convergent sequences are norm convergent), its dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024043.png" /> does not have the Dunford–Pettis property (because it contains complemented copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024044.png" />).

A [[Reflexive space|reflexive space]] does not have the Dunford–Pettis property unless it is finite-dimensional. The Grothendieck spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024047.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024048.png" /> (cf. [[Grothendieck space|Grothendieck space]]) also possess the Dunford–Pettis property (see [[#References|[a9]]], [[#References|[a10]]]).

A Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024049.png" /> is a Grothendieck space with the Dunford–Pettis property if and only if every weak-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024050.png" /> convergent sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024051.png" /> converges weakly and uniformly on weakly compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024052.png" />, if and only if every bounded [[Linear operator|linear operator]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024053.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024054.png" /> 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|Strong topology]]) implies [[Uniform convergence|uniform convergence]]. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024055.png" /> be a Grothendieck space with the Dunford–Pettis property. Then:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024056.png" /> does not have a Schauder decomposition, or equivalently, if a sequence of projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024058.png" /> converges weakly to the identity operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024059.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024060.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024061.png" /> sufficiently large;

2) if the Cesáro mean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024062.png" /> of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024063.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024064.png" /> converges strongly, then it converges uniformly;

3) all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024065.png" />-semi-groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024066.png" /> are norm-continuous (see [[#References|[a9]]], [[#References|[a10]]]);

4) all strongly continuous cosine operator functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024067.png" /> are norm-continuous [[#References|[a11]]];

5) for general ergodic systems on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024068.png" />, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024069.png" />-semi-groups and cosine operator functions, strong ergodicity implies uniform ergodicity (see [[#References|[a12]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bourgain,   "On the Dunford–Pettis property"  ''Proc. Amer. Math. Soc.'' , '''81'''  (1981)  pp. 265–272</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Bourgain,   "New Banach space properties of the disc algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024070.png" />"  ''Acta Math.'' , '''152'''  (1984)  pp. 1–48</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Bourgain,   "The Dunford–Pettis property for the ball-algebras, the polydisc algebra, and the Sobolev spaces"  ''Studia Math.'' , '''77'''  (1984)  pp. 245–253</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Diestel,   "A survey or results related to the Dunford–Pettis property" , ''Contemp. Math.'' , '''2''' , Amer. Math. Soc.  (1980)  pp. 15–60</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N. Dunford,   B.J. Pettis,   "Linear operations on summable functions"  ''Trans. Amer. Math. Soc.'' , '''47'''  (1940)  pp. 323–392</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators" , '''I. General theory''' , Wiley, reprint  (1988)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Grothendieck,   "Sur les applications linéaires faiblement compactes d'espaces de type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024071.png" />"  ''Canad. J. Math.'' , '''5'''  (1953)  pp. 129–173</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Talagrand,   "La propriété de Dunford–Pettis dans <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024072.png" /> et <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024073.png" />"  ''Israel J. Math.'' , '''44'''  (1983)  pp. 317–321</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  H.P. Lotz,   "Tauberian theorems for operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024074.png" /> and similar spaces" , ''Functional Analysis III. Surveys and Recent Results'' , North-Holland  (1984)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  H.P. Lotz,   "Uniform convergence of operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110240/d11024075.png" /> and similar spaces"  ''Math. Z.'' , '''190'''  (1985)  pp. 207–220</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  S.-Y. Shaw,   "Asymptotic behavior of pseudoresolvents on some Grothendieck spaces"  ''Publ. RIMS Kyoto Univ.'' , '''24'''  (1988)  pp. 277–282</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  S.-Y. Shaw,   "Uniform convergence of ergodic limits and approximate solutions"  ''Proc. Amer. Math. Soc.'' , '''114'''  (1992)  pp. 405–411</TD></TR></table>