Deviation from weak Banach–Saks property for countable direct sums

Andrzej Kryczka

Annales UMCS, Mathematica (2015)

  • Volume: 68, Issue: 2, page 51-58
  • ISSN: 2083-7402

Abstract

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We introduce a seminorm for bounded linear operators between Banach spaces that shows the deviation from the weak Banach-Saks property. We prove that if (Xν) is a sequence of Banach spaces and a Banach sequence lattice E has the Banach-Saks property, then the deviation from the weak Banach-Saks property of an operator of a certain class between direct sums E(Xν) is equal to the supremum of such deviations attained on the coordinates Xν. This is a quantitative version for operators of the result for the Köthe- Bochner sequence spaces E(X) that if E has the Banach-Saks property, then E(X) has the weak Banach-Saks property if and only if so has X.

How to cite

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Andrzej Kryczka. "Deviation from weak Banach–Saks property for countable direct sums." Annales UMCS, Mathematica 68.2 (2015): 51-58. <http://eudml.org/doc/269962>.

@article{AndrzejKryczka2015,
abstract = {We introduce a seminorm for bounded linear operators between Banach spaces that shows the deviation from the weak Banach-Saks property. We prove that if (Xν) is a sequence of Banach spaces and a Banach sequence lattice E has the Banach-Saks property, then the deviation from the weak Banach-Saks property of an operator of a certain class between direct sums E(Xν) is equal to the supremum of such deviations attained on the coordinates Xν. This is a quantitative version for operators of the result for the Köthe- Bochner sequence spaces E(X) that if E has the Banach-Saks property, then E(X) has the weak Banach-Saks property if and only if so has X.},
author = {Andrzej Kryczka},
journal = {Annales UMCS, Mathematica},
keywords = {Weak Banach-Saks property; K¨othe-Bochner space; direct sum; weak Banach-Saks property; Köthe-Bochner space},
language = {eng},
number = {2},
pages = {51-58},
title = {Deviation from weak Banach–Saks property for countable direct sums},
url = {http://eudml.org/doc/269962},
volume = {68},
year = {2015},
}

TY - JOUR
AU - Andrzej Kryczka
TI - Deviation from weak Banach–Saks property for countable direct sums
JO - Annales UMCS, Mathematica
PY - 2015
VL - 68
IS - 2
SP - 51
EP - 58
AB - We introduce a seminorm for bounded linear operators between Banach spaces that shows the deviation from the weak Banach-Saks property. We prove that if (Xν) is a sequence of Banach spaces and a Banach sequence lattice E has the Banach-Saks property, then the deviation from the weak Banach-Saks property of an operator of a certain class between direct sums E(Xν) is equal to the supremum of such deviations attained on the coordinates Xν. This is a quantitative version for operators of the result for the Köthe- Bochner sequence spaces E(X) that if E has the Banach-Saks property, then E(X) has the weak Banach-Saks property if and only if so has X.
LA - eng
KW - Weak Banach-Saks property; K¨othe-Bochner space; direct sum; weak Banach-Saks property; Köthe-Bochner space
UR - http://eudml.org/doc/269962
ER -

References

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  5. [5] Krassowska, D., Płuciennik, R., A note on property (H) in Köthe-Bochner sequence spaces, Math. Japon. 46 (1997), 407-412. Zbl0911.46003
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  8. [8] Kryczka, A., Mean separations in Banach spaces under abstract interpolation and extrapolation, J. Math. Anal. Appl. 407 (2013), 281-289.[WoS] Zbl06408407
  9. [9] Lin, P.-K., Köthe-Bochner function spaces, Birkhäuser Boston, Inc., Boston, MA, 2004. Zbl1054.46003
  10. [10] Lindenstrauss, J., Tzafriri, L., Classical Banach spaces. II. Function spaces, Springer- Verlag, Berlin-New York, 1979. Zbl0403.46022
  11. [11] Mastyło, M., Interpolation spaces not containing l1, J. Math. Pures Appl. 68 (1989), 153-162. Zbl0632.46066
  12. [12] Partington, J. R., On the Banach-Saks property, Math. Proc. Cambridge Philos. Soc. 82 (1977), 369-374. Zbl0368.46018
  13. [13] Rosenthal, H. P., Weakly independent sequences and the Banach-Saks property, Bull. London Math. Soc. 8 (1976), 22-24. 
  14. [14] Szlenk, W., Sur les suites faiblement convergentes dans l’espace L, Studia Math. 25 (1965), 337-341. Zbl0131.11505

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