Dunford Pettis property

The Dunford-Pettis property (according to N. Dunford and BJ Pettis ) is a property of Banach spaces considered in the mathematical sub-area of functional analysis .

definition

The following definition goes back to A. Grothendieck (1953):

A Banach space has the Dunford-Pettis property if for every Banach space every weakly compact linear operator is already complete . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X\rightarrow Y}$

After the English term "Dunford-Pettis-Property" one uses the abbreviation DPP and says briefly, have or is DPP. ${\displaystyle X}$

Examples

• The sequence spaces , and have the Dunford-Pettis property, but the sequence spaces do not. ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{\infty }}$ ${\displaystyle \ell ^{p},1<p<\infty }$
• If L ¹ is a finite measure space , then L ^{1 has} the Dunford-Pettis property. That this is the case has previously been proven by N. Dunford and BJ Pettis and was the motivation for Grothendieck to give the name. ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle (\Omega ,\Sigma ,\mu )}$
• If a compact Hausdorff space , then the Banach space of continuous functions has the Dunford-Pettis property, as was proven by Grothendieck. ${\displaystyle K}$ ${\displaystyle C(K)}$ ${\displaystyle K\to \mathbb {C} }$
• No infinite-dimensional reflexive Banach space has the Dunford-Pettis property.

A characterization

The following statements are equivalent for a Banach space : ${\displaystyle X}$

• ${\displaystyle X}$ has the Dunford Pettis property.
• If there is a sequence in with a weak limit value and a sequence in the dual space with a weak limit value , then applies to . ${\displaystyle (x_{n})_{n}}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle (f_{n})_{n}}$ ${\displaystyle X'}$ ${\displaystyle f}$ ${\displaystyle f_{n}(x_{n})\to f(x)}$ ${\displaystyle n\to \infty }$
• If there is a sequence in with a weak limit value and a sequence in the dual space with a weak limit value , then applies to . ${\displaystyle (x_{n})_{n}}$ ${\displaystyle X}$ ${\displaystyle 0}$ ${\displaystyle (f_{n})_{n}}$ ${\displaystyle X'}$ ${\displaystyle 0}$ ${\displaystyle f_{n}(x_{n})\to 0}$ ${\displaystyle n\to \infty }$

properties

If the dual space of the Banach space has the Dunford-Pettis property, so too . ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle X}$

Since the commutative C * algebra is of the form with a compact Hausdorff space (see Gelfand-Neumark theorem ), it has the Dunford-Pettis property according to the Grothendieck theorem mentioned under the examples. Since and (see article sequence space ), it follows that also and have the Dunford-Pettis property. ${\displaystyle \ell ^{\infty }}$ ${\displaystyle C(K)}$ ${\displaystyle K}$ ${\displaystyle \ell ^{\infty }}$ ${\displaystyle (c_{0})'\cong \ell ^{1}}$ ${\displaystyle (\ell ^{1})'\cong \ell ^{\infty }}$ ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{1}}$

By definition, all weakly compact operators on spaces with the Dunford-Pettis property are complete, but the converse does not have to hold. For example, has the Dunford-Pettis property and the identity is complete, because because of the Schur property , weakly compact sets are already norm-compact. but not weak-compact, otherwise the unit sphere would already weak-compact and would reflexively , but that is not the case. ${\displaystyle \ell ^{1}}$ ${\displaystyle \mathrm {id} _{\ell ^{1}}}$ ${\displaystyle \mathrm {id} _{\ell ^{1}}}$ ${\displaystyle \ell ^{1}}$

swell

• Robert E. Megginson: An Introduction to Banach Space Theory . Springer, New York 1998, ISBN 0-387-98431-3 .
• Joseph Diestel: Sequences and Series in Banach Spaces . Springer, New York, Berlin, Heidelberg, Tokyo 1984, ISBN 0-387-90859-5 .