On the class of order Dunford-Pettis operators

Khalid Bouras; Abdelmonaim El Kaddouri; Jawad H'michane; Mohammed Moussa

Mathematica Bohemica (2013)

  • Volume: 138, Issue: 3, page 289-297
  • ISSN: 0862-7959

Abstract

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We characterize Banach lattices E and F on which the adjoint of each operator from E into F which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if E and F are two Banach lattices then each order Dunford-Pettis and weak Dunford-Pettis operator T from E into F has an adjoint Dunford-Pettis operator T ' from F ' into E ' if, and only if, the norm of E ' is order continuous or F ' has the Schur property. As a consequence we show that, if E and F are two Banach lattices such that E or F has the Dunford-Pettis property, then each order Dunford-Pettis operator T from E into F has an adjoint T ' : F ' ⟶ E ' which is Dunford-Pettis if, and only if, the norm of E ' is order continuous or F ' has the Schur property.

How to cite

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Bouras, Khalid, et al. "On the class of order Dunford-Pettis operators." Mathematica Bohemica 138.3 (2013): 289-297. <http://eudml.org/doc/260579>.

@article{Bouras2013,
abstract = {We characterize Banach lattices $E$ and $F$ on which the adjoint of each operator from $E$ into $F$ which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if $E$ and $F$ are two Banach lattices then each order Dunford-Pettis and weak Dunford-Pettis operator $T$ from $E$ into $F$ has an adjoint Dunford-Pettis operator $T^\{\prime \}$ from $F^\{\prime \}$ into $E^\{\prime \}$ if, and only if, the norm of $E^\{\prime \}$ is order continuous or $F^\{\prime \}$ has the Schur property. As a consequence we show that, if $E$ and $F$ are two Banach lattices such that $E$ or $F$ has the Dunford-Pettis property, then each order Dunford-Pettis operator $T$ from $E$ into $F$ has an adjoint $T^\{\prime \}\colon F^\{\prime \}\longrightarrow E^\{\prime \}$ which is Dunford-Pettis if, and only if, the norm of $E^\{\prime \}$ is order continuous or $F^\{\prime \}$ has the Schur property.},
author = {Bouras, Khalid, El Kaddouri, Abdelmonaim, H'michane, Jawad, Moussa, Mohammed},
journal = {Mathematica Bohemica},
keywords = {Dunford-Pettis operator; weak Dunford-Pettis operator; order Dunford-Pettis operator; order continuous norm; Schur property; Dunford-Pettis operator; weak Dunford-Pettis operator; order Dunford-Pettis operator; order continuous norm; Schur property},
language = {eng},
number = {3},
pages = {289-297},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the class of order Dunford-Pettis operators},
url = {http://eudml.org/doc/260579},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Bouras, Khalid
AU - El Kaddouri, Abdelmonaim
AU - H'michane, Jawad
AU - Moussa, Mohammed
TI - On the class of order Dunford-Pettis operators
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 3
SP - 289
EP - 297
AB - We characterize Banach lattices $E$ and $F$ on which the adjoint of each operator from $E$ into $F$ which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if $E$ and $F$ are two Banach lattices then each order Dunford-Pettis and weak Dunford-Pettis operator $T$ from $E$ into $F$ has an adjoint Dunford-Pettis operator $T^{\prime }$ from $F^{\prime }$ into $E^{\prime }$ if, and only if, the norm of $E^{\prime }$ is order continuous or $F^{\prime }$ has the Schur property. As a consequence we show that, if $E$ and $F$ are two Banach lattices such that $E$ or $F$ has the Dunford-Pettis property, then each order Dunford-Pettis operator $T$ from $E$ into $F$ has an adjoint $T^{\prime }\colon F^{\prime }\longrightarrow E^{\prime }$ which is Dunford-Pettis if, and only if, the norm of $E^{\prime }$ is order continuous or $F^{\prime }$ has the Schur property.
LA - eng
KW - Dunford-Pettis operator; weak Dunford-Pettis operator; order Dunford-Pettis operator; order continuous norm; Schur property; Dunford-Pettis operator; weak Dunford-Pettis operator; order Dunford-Pettis operator; order continuous norm; Schur property
UR - http://eudml.org/doc/260579
ER -

References

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  2. Andrews, K. T., 10.1007/BF01406706, Math. Ann. 241 (1979), 35-41. (1979) Zbl0398.46025MR0531148DOI10.1007/BF01406706
  3. Aqzzouz, B., Bouras, K., Weak and almost Dunford-Pettis operators on Banach lattices, Demonstr. Math. 46 165-179 (2013). (2013) MR3075506
  4. Aqzzouz, B., Bouras, K., Dunford-Pettis sets in Banach lattices, Acta Math. Univ. Comen., New Ser. 81 185-196 (2012). (2012) MR2975284
  5. Aqzzouz, B., Bouras, K., Moussa, M., Duality property for positive weak Dunford-Pettis operators, Int. J. Math. Math. Sci. 2011, Article ID 609287 12 p (2011). (2011) Zbl1262.47057MR2821970
  6. Dodds, P. G., Fremlin, D. H., 10.1007/BF02760610, Isr. J. Math. 34 (1979), 287-320. (1979) MR0570888DOI10.1007/BF02760610
  7. Meyer-Nieberg, P., Banach Lattices, Universitext. Springer, Berlin (1991). (1991) Zbl0743.46015MR1128093

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