In this paper, a new class of Banach spaces, termed as Banach spaces with property (MB), will be introduced. It is stated that a space X has property (MB) if every V -subset of X* is an L-subset of X* . We describe those spaces which have property (MB) . Also, we show that if a Banach space X has property (MB) and Banach space Y does not contain , then every operator is completely continuous.
References
[1]
Saab, P. and Smith, B. (1992) Spaces on Which Every Unconditionally Converging Operators Are Weakly Completely Continuous. Rocky Mountain Journal of Mathematics, 22, 1001-1009. https://doi.org/10.1216/rmjm/1181072710
[2]
Bator, E., Lewis, P. and Ochoa, J. (1998) Evaluation Maps, Restriction Maps, and Compactness. Colloquium Mathematicum, 78, 1-17.
[3]
Diestel, J. (1984) Sequences and Series in Banach Spaces. Grad Texts in Math, No. 92, Springer-Verlag, Berlin. https://doi.org/10.1007/978-1-4612-5200-9
[4]
Dunford, N. and Schwartz, J.T. (1958) Linear Operators, Part I: General Theory. Wiley-Interscience, New York.
[5]
Bator, E.M. (1987) Remarks on Completely Continuous Operators. Bulletin of the Polish Academy of Sciences Mathematics, 37, 409-413.
[6]
Emmanuele, G. (1986) A Dual Characterization of Banach Spaces not Containing . Bulletin of the Polish Academy of Sciences Mathematics, 34, 155-160.
[7]
Emmanuele, G. (1991) On the Reciprocal Dunford-Pettis Property in Projective Tensor Products. Mathematical Proceedings of the Cambridge Philosophical Society, 109, 161-166. https://doi.org/10.1017/S0305004100069632
[8]
Diestel, J. (1980) A Survey of Results Related to the Dunford-Pettis Property. Contemporary Math., 2, 15-60. https://doi.org/10.1090/conm/002/621850
[9]
Pelczyinski, A. (1962) Banach Spaces on Which Every Unconditionally Converging Operator Is Weakly Compact. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 10, 641-648.
[10]
Soyabs, D. (2012) The (D) Property in Banach Spaces. Abstract and Applied Analysis.
[11]
Saab, E. and Saab, P. (1993) Extensions of Some Classes of Operators and Applications. Rocky Mountain Journal of Mathematics, 23, 319-337.
[12]
Rosenthal, H. (1977) Pointwise Compact Subsets of the First Baire Class. American Journal of Mathematics, 99, 362-377. https://doi.org/10.2307/2373824
[13]
Pelczyinski, A. and Semadeni, Z. (1959) Spaces of Continuous Functions (III). Studia Mathematica, 18, 211-222.
![\"\"](\"Edit_a00bf6c8-657c-4281-b034-8694d5db97d3.bmp\")
, then every operator ![\"\"](\"Edit_ca0107c8-f401-483e-8e44-466609dc0b87.bmp\")
is completely continuous.
