Deviation from weak Banach–Saks property for countable direct sums

Andrzej Kryczka

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.

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References


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

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

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

Erdös, P., Magidor, M., A note on regular methods of summability and the Banach– Saks property, Proc. Amer. Math. Soc. 59 (1976), 232–234.

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

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.

Kryczka, A., Alternate signs Banach–Saks property and real interpolation of operators, Proc. Amer. Math. Soc. 136 (2008), 3529–3537.

Kryczka, A., Mean separations in Banach spaces under abstract interpolation and extrapolation, J. Math. Anal. Appl. 407 (2013), 281–289.

Lin, P.-K., Köthe–Bochner function spaces, Birkhäuser Boston, Inc., Boston, MA, 2004.

Lindenstrauss, J., Tzafriri, L., Classical Banach spaces. II. Function spaces, Springer- Verlag, Berlin–New York, 1979.

Mastyło, M., Interpolation spaces not containing l1, J. Math. Pures Appl. 68 (1989), 153–162.

Partington, J. R., On the Banach–Saks property, Math. Proc. Cambridge Philos. Soc. 82 (1977), 369–374.

Rosenthal, H. P., Weakly independent sequences and the Banach–Saks property, Bull. London Math. Soc. 8 (1976), 22–24.

Szlenk, W., Sur les suites faiblement convergentes dans l’espace L, Studia Math. 25 (1965), 337–341.




DOI: http://dx.doi.org/10.17951/a.2014.68.2.51
Date of publication: 2015-05-23 16:29:45
Date of submission: 2015-05-09 13:33:31


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