## Annals of Functional Analysis

### On the $p$-Schur property of Banach spaces

#### Abstract

We introduce the notion of the $p$-Schur property ($1\leq p\leq\infty$) as a generalization of the Schur property of Banach spaces, and then we present a number of basic properties and some examples. We also study its relation with some geometric properties of Banach spaces, such as the Gelfand–Phillips property. Moreover, we verify some necessary and sufficient conditions for the $p$-Schur property of some closed subspaces of operator spaces.

#### Article information

Source
Ann. Funct. Anal. (2017), 14 pages.

Dates
Accepted: 8 March 2017
First available in Project Euclid: 5 October 2017

https://projecteuclid.org/euclid.afa/1507169077

Digital Object Identifier
doi:10.1215/20088752-2017-0033

#### Citation

Dehghani, Mohammad B.; Moshtaghioun, S. Mohammad. On the p -Schur property of Banach spaces. Ann. Funct. Anal., advance publication, 5 October 2017. doi:10.1215/20088752-2017-0033. https://projecteuclid.org/euclid.afa/1507169077

#### References

• [1] F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Grad. Texts in Math. 233, Springer, New York, 2006.
• [2] J. Bourgain and J. Diestel, Limited operators and strict cosingularity, Math. Nachr. 119 (1984), 55–58.
• [3] S. W. Brown, Weak sequential convergence in the dual of an algebra of compact operators, J. Operator Theory 33 (1995), no. 1, 33–42.
• [4] J. Castillo and F. Sanchez, Dunford–Pettis-like properties of continuous vector function spaces, Rev. Mat. Complut. 6 (1993), no. 1, 43–59.
• [5] D. Chen, J. A. Chávez-Domínguez, and L. Li, Unconditionally $p$-converging operators and Dunford–Pettis property of order $p$, preprint, arXiv:1607.02161v1 [math.FA].
• [6] A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland, Amsterdam, 1993.
• [7] R. Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monogr. Surv. Pure Appl. Math. 64, Wiley, New York, 1993.
• [8] J. Diestel, “A survey of results related to the Dunford–Pettis property” in Integration, Topology, and Geometry in Linear Spaces (Chapel Hill, NC, 1979), Contemp. Math. 2, Amer. Math. Soc., Providence, 1980, 15–60.
• [9] L. Drewnowski, On Banach spaces with the Gelfand–Phillips property, Math. Z. 193 (1986), no. 3, 405–411.
• [10] G. Emmanuele, On Banach spaces with the Gelfand–Phillips property, III, J. Math. Pures Appl. (9) 72 (1993), no. 3, 327–333.
• [11] J. H. Fourie and E. D. Zeekoei, $\mathit{DP}^{*}$-properties of order $p$ on Banach spaces, Quaest. Math. 37 (2014), no. 3, 349–358.
• [12] O. F. K. Kalenda and J. Spurný, On a difference between quantitative weak sequential completeness and the quantitative Schur property, Proc. Amer. Math. Soc. 140 (2012), no. 10, 3435–3444.
• [13] S. M. Moshtaghioun and J. Zafarani, Weak sequential convergence in the dual of operator ideals, J. Operator Theory 49 (2003), no. 1, 143–151.
• [14] A. Pelczynski, Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Pol. Sci. 10 (1962), 641–648.
• [15] R. A. Ryan, The Dunford–Pettis property and projective tensor products, Bull. Pol. Acad. Sci. Math. 35 (1987), no. 11-12, 785–792.
• [16] R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Monogr. Math., Springer, London, 2002.
• [17] M. Salimi and S. M. Moshtaghioun, The Gelfand–Phillips property in closed subspaces of some operator spaces, Banach J. Math. Anal. 5 (2011), no. 2, 84–92.
• [18] D. P. Sinha and A. K. Karn, Compact operators whose adjoints factor through subspaces of $\ell_{p}$, Studia Math. 150 (2002), no. 1, 17–33.
• [19] A. Ülger, Subspaces and subalgebras of $K(H)$ whose duals have the Schur property, J. Operator Theory 37 (1997), no. 2, 371–378.