# Deviation from weak Banach–Saks property for countable direct sums

Annales UMCS, Mathematica (2015)

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

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topAndrzej 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

top- [1] Banach, S., Saks, S., Sur la convergence forte dans les champs Lp, Studia Math. 2 (1930), 51-57. Zbl56.0932.01
- [2] Beauzamy, B., Banach-Saks properties and spreading models, Math. Scand. 44 (1979), 357-384. Zbl0427.46007
- [3] Brunel, A., Sucheston, L., On B-convex Banach spaces, Math. Systems Theory 7 (1974), 294-299. Zbl0323.46018
- [4] Erdös, P., Magidor, M., A note on regular methods of summability and the Banach- Saks property, Proc. Amer. Math. Soc. 59 (1976), 232-234. Zbl0355.40007
- [5] Krassowska, D., Płuciennik, R., A note on property (H) in Köthe-Bochner sequence spaces, Math. Japon. 46 (1997), 407-412. Zbl0911.46003
- [6] Krein, S. G., Petunin, Yu. I., Semenov, E. M., Interpolation of linear operators, Translations of Mathematical Monographs, 54. American Mathematical Society, Providence, R.I., 1982.
- [7] Kryczka, A., Alternate signs Banach-Saks property and real interpolation of operators, Proc. Amer. Math. Soc. 136 (2008), 3529-3537.[WoS] Zbl1160.46014
- [8] Kryczka, A., Mean separations in Banach spaces under abstract interpolation and extrapolation, J. Math. Anal. Appl. 407 (2013), 281-289.[WoS] Zbl06408407
- [9] Lin, P.-K., Köthe-Bochner function spaces, Birkhäuser Boston, Inc., Boston, MA, 2004. Zbl1054.46003
- [10] Lindenstrauss, J., Tzafriri, L., Classical Banach spaces. II. Function spaces, Springer- Verlag, Berlin-New York, 1979. Zbl0403.46022
- [11] Mastyło, M., Interpolation spaces not containing l1, J. Math. Pures Appl. 68 (1989), 153-162. Zbl0632.46066
- [12] Partington, J. R., On the Banach-Saks property, Math. Proc. Cambridge Philos. Soc. 82 (1977), 369-374. Zbl0368.46018
- [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. Zbl0131.11505

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