Abstract
Let X and Y be two Banach spaces. A bounded operator \(T: X \longrightarrow Y\) is said to be a BS-compact operator whenever T sends Banach-Saks subsets of X onto norm compact sets of Y. In this paper, our central focus is upon introducing the class of almost BS-compact operators. The paper rests essentially on two parts. The first is devoted to the connection of this new class of operators with classical notions of operators, such as BS-compact operators, AM-compact operators, and Dunfort-Pettis operators. The second part is dedicated to the domination problem within the framework of (almost) BS-compact operators.
Similar content being viewed by others
References
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Berlin (2006)
Aliprantis, C.D., Burkinshaw, O.: Positive compact operators on Banach lattices. Math. Z. 174, 289–298 (1980)
Aliprantis, C., D., Burkinshaw, O. : On weakly compact operators on Banach lattices. Proc. Amer. Math. Soc. 83,(1981)
Aliprantis, C., D., Burkinshaw, O. : Dunford-Pettis operators on Banach lattices. Trans. Amer. Math. Soc. (274 (1982).)
Aqzzouz, B., Nouira, R., Zraoula, L.: Compactness properties for operators dominated by AM-compact operators. Trans. Amer. Math. Soc. 153(4), 1151–1157 (2007)
Baernstein, A.: On reflexivity and summability II. Studia Math. 42, 91–94 (1972)
Baklouti, H., Hajji, M.: Domination problem on Banach lattices and almost weak compactness. Positivity. 19, 797–805 (2015)
Baklouti, H., Hajji, M.: Schur operators and domination problem. Positivity. 21, 35–48 (2017)
Baklouti, H., Hajji, M.: Disjointly improjective operators and domination problem. Indagationes Mathematicae 28, 1175–1182 (2017)
Banach, S., Saks, S.: Sur la convergence forte dans les champs \(L_p.\) Studia. Math. 2, 51–57 (1930)
Beauzamy, B.: Propriété de Banach–Saks, ibid. 66, 227-235 (1980)
Bouras, K., Lhaimer, D., Moussa, M.: On the class of almost L-weakly and almost M-weakly compact operators. Positivity 22, 1433–1443 (2018)
Brunel, A., Sucheston, L.: On J-convexity and some ergodic super-properties of Banach spaces. Proc. Amer. Math. Soc. 204, 79–90 (1975)
Diestel, J.: Sequences and series in Banach spaces. Springer-Verlag, New York (1984)
Compact operators in Banach lattice: Dodds, P, G., Fremlin, D, H. Isr. J. Math. 34, 287–320 (1979)
Erdos, P., Magidor, M.: A note on regular methods of summability and the Banach-Saks property. Proc. Amer. Math. Soc. 59, 232–234 (1976)
Fabian, M., Habala, P., HAjek, P., Montesinos, V., Zizler, V. : Banach space theory: basis for linear and nonlinear analysis. Springer-Verlag, New York (2011)
Powers of operators dominated by strictly singular operators: Flores, J., Hernandez, F, L., Tradacete, P. Proc. Q. J. Math. Soc. 59, 321–334 (2008)
Ghoussoub, N., Johnson, W.B.: Factoring operators through Banach lattices not containing \(C(0, 1).\). Math. Z. 194, 153–171 (1987)
Jarosz, K.: Function Spaces: Proceedings of the Third Conference on Function Spaces, pp. 19–23. , Southern Illinois University at Edwardsville, May (1998)
Kalton, N., Saab, P.: Ideal properties of regular operators between Banach lattices. Illinois J. Math. 29, 382–400 (1985)
G, Groenewegen.: On spaces of Banach lattice valued functions and measures, PhD Thesis, Nijmegen University, 1982
Larsen, R.: Functional analysis: An introduction. Marcel Dekker Inc, New York (1973)
Lindenstrauss, J., Tzafriri, L.: Classical Banach space II. Function Spaces, Springer, New york (1979)
Núñez, C.: Characterization of Banach Spaces of Continuous Vector Valued Functions with The Weak Banach-Saks Property. Illinois Journal of Mathematics. 33(1), 27–41 (1989)
Lopez-Abad, J., Ruiz, C., Tradacete, P.: The convex hull of a Banach-Saks set. Journal of Functional Analysis. 266(4), 2251–2280 (2014)
Meyer-Nieberg, P.: Uber Klassen Schwach Kompakter Operatoren in Banachverbanden. Math. Z. 138, 145–159 (1974)
Meyer-Nieberg, P.: Banach lattices. Springer-Verlag, Berlin, Heidelberg, New York (1991)
Nishiura, T., Waterman, D.: Reflexivity and summability. Studia Math. 23, 53–57 (1963)
Szlenk, W.: Sur les suites faiblement convergentes dans l’espace l. Studia Math. 25, 337–341 (1965)
Tradacete, P.: Factorization and domination properties of operators on Banach Latices. Universidad Complutense de Madrid (2010).. (Phd thesis)
Weis, L.: Banach lattices with the subsequence splitting property. Proc. Am. Math. Soc. 105, 87–96 (1989)
Acknowledgements
The author would like to express their cordial gratitude to the referee for valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.