Geodesic
Deviation from weak Banach–Saks property for countable direct sums
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 68 (2014) no. 2.

Voir la notice de l'article provenant de la source Library of Science

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. 
@article{AUM_2014_68_2_a0,
     author = {Kryczka, Andrzej},
     title = {Deviation from weak {Banach{\textendash}Saks} property for countable direct sums},
     journal = {Annales Universitatis Mariae Curie-Sk{\l}odowska. Mathematica },
     publisher = {mathdoc},
     volume = {68},
     number = {2},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUM_2014_68_2_a0/}
}
TY  - JOUR
AU  - Kryczka, Andrzej
TI  - Deviation from weak Banach–Saks property for countable direct sums
JO  - Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
PY  - 2014
VL  - 68
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AUM_2014_68_2_a0/
LA  - en
ID  - AUM_2014_68_2_a0
ER  - 
%0 Journal Article
%A Kryczka, Andrzej
%T Deviation from weak Banach–Saks property for countable direct sums
%J Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
%D 2014
%V 68
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AUM_2014_68_2_a0/
%G en
%F AUM_2014_68_2_a0
Kryczka, Andrzej. Deviation from weak Banach–Saks property for countable direct sums. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 68 (2014) no. 2. http://geodesic.mathdoc.fr/item/AUM_2014_68_2_a0/

[1] Banach, S., Saks, S., Sur la convergence forte dans les champs Lp, Studia Math. 2 (1930), 51–57.

[2] Beauzamy, B., Banach–Saks properties and spreading models, Math. Scand. 44 (1979), 357–384.

[3] Brunel, A., Sucheston, L., On B-convex Banach spaces, Math. Systems Theory 7 (1974), 294–299.

[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.

[5] Krassowska, D., Płuciennik, R., A note on property (H) in Köthe–Bochner sequence spaces, Math. Japon. 46 (1997), 407–412.

[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.

[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.