Dunford-Pettis operator

In their classic 1940 paper [a5], N. Dunford and B.J. Pettis (with a bit of help from R.S. Phillips, [a8]) showed that if $T : L ^ { 1 } \rightarrow X$ is a weakly compact operator (cf. Dunford–Pettis property; Grothendieck space) acting on a space $L^1$ of Lebesgue-integrable functions, then $T$ is completely continuous (cf. also Completely-continuous operator); hence, if $T : X \rightarrow L ^ { 1 }$ and $S : L ^ { 1 } \rightarrow Y$ are weakly compact, then $S T : X \rightarrow Y$ is compact. Here, an operator is (weakly) compact if it takes bounded sets into (weakly) compact sets and completely continuous if it takes weakly compact sets into norm-compact sets. See also Dunford–Pettis property.

The Dunford–Pettis result was recognized by A. Grothendieck for what it was and, in his seminal 1953 paper [a6], he isolated several isomorphic invariants inspired by the work of Dunford and Pettis. In particular, he said that a Banach space $X$ has the Dunford–Pettis property if for any Banach space $Y$, any weakly compact operator $T : X \rightarrow Y$ is completely continuous, while $X$ has the reciprocal Dunford–Pettis property if regardless of the Banach space $Y$, the weak compactness of a linear operator $T : X \rightarrow Y$ is ensured by its complete continuity. These definitions were the first formulations in terms of how classes of operators on a space relate to each other and a clear indication of the impact homological thinking was having on Grothendieck and, through him, on functional analysis.

Grothendieck did more than define the properties; he showed that for any compact Hausdorff space $K$, the space $C ( K )$ of continuous scalar-valued functions on $K$ enjoys both the Dunford–Pettis property and the reciprocal Dunford–Pettis property. Soon after, Grothendieck used ideas related to the Dunford–Pettis property to show that for a finite measure $\mu$, any linear subspace of $L ^ { \infty } ( \mu )$ that is closed in $L ^ { p } ( \mu )$ ($1 \leq p < \infty$) is finite-dimensional. After Grothendieck, efforts at adding new, significant examples of spaces with the Dunford–Pettis property met with little success; in the late 1970s, J. Elton and E. Odell discovered that any infinite-dimensional Banach space contains either a copy of $c_0$, $\mathbf{l}_{1}$ or a subspace without the Dunford–Pettis property. Interest in the serious study of the Dunford–Pettis property was renewed, although new and different examples of spaces with the Dunford–Pettis property were still elusive.

The logjam was broken in 1983, when J. Bourgain [a1] showed that the poly-disc algebras, poly-ball algebras and the spaces of continuously differentiable functions all enjoy the Dunford–Pettis property; Bourgain showed much more: the aforementioned spaces and all their duals enjoy the Dunford–Pettis property. Bourgain's work was to lead to a rash of new, interesting examples and techniques, Bourgain algebras were borne (cf. [a11]) and the already tight relations between Banach space theory and harmonic analysis were further solidified.

References

[a1]	J. Bourgain, "The Dunford–Pettis property for the ball-algebras, the polydisc algebras and the Soboler spaces" Studia Math. , 77 (1984) pp. 245–253
[a2]	J.A. Cima, R.M. Timoney, "The Dunford–Pettis property for certain planar uniform algebras" Michigan Math. J. , 34 (1987) pp. 99–104
[a3]	J. Diestel, "A survey of results related to the Dunford–Pettis property" , Integration, Topology, and Geometry in Linear Spaces. Proc. Conf. Chapel Hill 1979 , Contemp. Math. , Amer. Math. Soc. (1980) pp. 15–60
[a4]	J. Diestel, J.J. Uhl Jr., "Vector Measures" , Surveys , 15 , Amer. Math. Soc. (1977)
[a5]	N. Dunford, B.J. Pettis, "Linear operations on summable functions" Trans. Amer. Math. Soc. , 47 (1940) pp. 323–390
[a6]	A. Grothendieck, "Sur les Applications linéaires faiblement compactes d'espaces du type $C ( K )$" Canad. J. Math. , 5 (1953) pp. 129–173
[a7]	E. Odell, "Applications of Ramsey theorems in Banach spaces" H.E. Lacey (ed.) , Notes in Banach Spaces , Austin Univ. Texas Press (1981)
[a8]	R.S. Phillips, "On linear transformations" Trans. Amer. Math. Soc. , 48 (1940) pp. 516–541
[a9]	S.F. Saccone, "Banach space properties of strongly tight uniform algebras" Studia Math. , 114 (1985) pp. 159–180
[a10]	P. Wojtaszczyk, "Banach spaces for analysts" , Studies Adv. Math. , 25 , Cambridge Univ. Press (1991)
[a11]	K. Yale, "Bourgain algebras" , Function spaces (Edwardsville, IL, 1990) , Lecture Notes Pure Appl. Math. , 136 , M. Dekker (1992) pp. 413–422