Title: On the Limited p-Schur Property of Some Operator Spaces
Author(s): M.B. Dehghani, S.M. Moshtaghioun, M. Dehghani
Pages: 50-61
Cite as:
M.B. Dehghani, S.M. Moshtaghioun, M. Dehghani, On the Limited p-Schur Property of Some Operator Spaces, Int. J. Anal. Appl., 16 (1) (2018), 50-61.

Abstract


We introduce and study the notion of limited $p$-Schur property ($1\leq p\leq\infty$) of Banach spaces. Also, we establish some necessary and sufficient conditions under which some operator spaces have the limited $p$-Schur property. In particular, we prove that if $X$ and $Y$ are two Banach spaces such that $X$ contains no copy of $\ell_1$ and $Y$ has the limited $p$-Schur property, then $K(X,Y)$ (the space of all compact operators from $X$ into $Y$) has the limited $p$-Schur property.

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References


  1. F. Albiac and N.J. Kalton, Topics in Banach Space Theory, Graduate Texts in Mathematics, 233, Springer, New York, 2006. Google Scholar

  2. J. Bourgain and J. Diestel, Limited operators and strict consingularity, Math. Nachr. 119 (1984) 55-58. Google Scholar

  3. J. Castillo and F. Sanchez, Dunford-Pettis-like properties of continuous vector function spaces, Rev. Mat. Univ. Complut. Madrid 6 (1993), no. 1, 43-59. Google Scholar

  4. D. Chen, J. Alejandro Chvez-Domnguez, and Li. Lei. Unconditionally p-converging operators and Dunford-Pettis Property of order p, arXiv preprint arXiv:1607.02161 (2016). Google Scholar

  5. A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Mathematics Studies, 176, North-Holland Publishing Co., Amsterdam, 1993. Google Scholar

  6. Mohammad B. Dehghani and S. Mohammad Moshtaghioun, On the p-Schur property of Banach spaces, Ann. Funct. Anal. (2017), 14 pages. Google Scholar

  7. M.B. Dehghani and S.M. Moshtaghioun, Limited p-converging operators and its relation with some geometric properties of Banach spaces, (2017), Submitted. Google Scholar

  8. J. Diestel, H. Jachowr and A. Tonge, Absolutely summing operators, Cambrigde University Press, 1995. Google Scholar

  9. G. Emmanuele, On relative compactness in K(X,Y ), J. Math. Anal. Appl. 379 (2013) 88-90. Google Scholar

  10. J. H. Fouire and E. D. Zeekoei, DP ∗ properties of order p on Banach spaces, Quaest. Math. 37 (2014), no. 3, 349-358. Google Scholar

  11. I. Ghencia and P. Lewis, The Dunford-Pettis property, the Gelfand-Phillips property and L-set, Colloq. Math. 1.6 (2006), 311-324. Google Scholar

  12. H. Jarchow, Locally convex spaces, B.G. Teubner, 1981. Google Scholar

  13. F. Mayoral, Compact sets of compact operators in absence of ‘ 1 , Proc. Amer. Math. Soc. 129 (2001), 7982. Google Scholar

  14. S.M. Moshtaghioun and J. Zafarani, Weak sequentional convergence in the dual of operator ideas, J. Oper. Theory 49 (2003), 143-151. Google Scholar

  15. A. Pelczynski, Banach Spaces in which every unconditionally converging operator is weakly compact, Bull. L’Acad. Polon. Sci. 10 no. 2, (1962), 641-648. Google Scholar

  16. R. Ryan,The Dunford-Pettis property and projective tensor products, Bull. Polish Acad. Sci. 35 no. 11-12, (1987), 785-792. Google Scholar

  17. T. Schlumprecht, Limited sets in Banach spaces, Ph. D. Dissertation, München, (1987). Google Scholar

  18. B. Tanbay, Direct sums and the Schur property, Turk. J. Math. 22 (1999), 349-354. Google Scholar