Deviation from weak Banach–Saks property for countable direct sums

Andrzej Kryczka

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2014)

  • Volume: 68, Issue: 2
  • ISSN: 0365-1029

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 (Xv) 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(Xv) is equal to the supremum of such deviations attained on the coordinates Xv. 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 Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 68.2 (2014): null. <http://eudml.org/doc/289842>.

@article{AndrzejKryczka2014,
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 (Xv) 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(Xv) is equal to the supremum of such deviations attained on the coordinates Xv. 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 Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {},
language = {eng},
number = {2},
pages = {null},
title = {Deviation from weak Banach–Saks property for countable direct sums},
url = {http://eudml.org/doc/289842},
volume = {68},
year = {2014},
}

TY - JOUR
AU - Andrzej Kryczka
TI - Deviation from weak Banach–Saks property for countable direct sums
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2014
VL - 68
IS - 2
SP - null
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 (Xv) 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(Xv) is equal to the supremum of such deviations attained on the coordinates Xv. 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 -
UR - http://eudml.org/doc/289842
ER -

References

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  8. Kryczka, A., Mean separations in Banach spaces under abstract interpolation and extrapolation, J. Math. Anal. Appl. 407 (2013), 281–289. 
  9. Lin, P.-K., Köthe–Bochner function spaces, Birkhäuser Boston, Inc., Boston, MA, 2004. 
  10. Lindenstrauss, J., Tzafriri, L., Classical Banach spaces. II. Function spaces, Springer- Verlag, Berlin–New York, 1979. 
  11. Mastyło, M., Interpolation spaces not containing l1, J. Math. Pures Appl. 68 (1989), 153–162. 
  12. Partington, J. R., On the Banach–Saks property, Math. Proc. Cambridge Philos. Soc. 82 (1977), 369–374. 
  13. Rosenthal, H. P., Weakly independent sequences and the Banach–Saks property, Bull. London Math. Soc. 8 (1976), 22–24. 
  14. Szlenk, W., Sur les suites faiblement convergentes dans l’espace L, Studia Math. 25 (1965), 337–341. 

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