## Annals of Functional Analysis

### A note on relative compactness in ${K(X,Y)}$

Ioana Ghenciu

#### Abstract

For Banach spaces $X$ and $Y$, let $K(X,Y)$ denote the space of all compact operators from $X$ to $Y$ endowed with the operator norm. We give sufficient conditions for subsets of $K(X,Y)$ to be relatively compact. We also give some necessary and sufficient conditions for the Dunford–Pettis relatively compact property of some spaces of operators.

#### Article information

Source
Ann. Funct. Anal. Volume 7, Number 3 (2016), 470-483.

Dates
Accepted: 26 January 2016
First available in Project Euclid: 19 July 2016

https://projecteuclid.org/euclid.afa/1468952109

Digital Object Identifier
doi:10.1215/20088752-3624814

Mathematical Reviews number (MathSciNet)
MR3528378

Zentralblatt MATH identifier
06621453

#### Citation

Ghenciu, Ioana. A note on relative compactness in K ( X , Y ) . Ann. Funct. Anal. 7 (2016), no. 3, 470--483. doi:10.1215/20088752-3624814. https://projecteuclid.org/euclid.afa/1468952109.

#### References

• [1] J. Bourgain, New Classes of $\mathcal{L}_{p}$-spaces, Lecture Notes in Math. 889, Springer, Berlin, 1981.
• [2] J. Bourgain and J. Diestel, Limited operators and strict cosingularity, Math. Nachrichten 119 (1984), 55–58.
• [3] J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math. 92, Springer, New York, 1984.
• [4] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, 1977.
• [5] L. Drewnowski and G. Emmanuele, On Banach spaces with the Gelfand–Phillips property, II, Rend. Circ. Mat. Palermo (2) 38 (1989), no. 3, 377–391.
• [6] G. Emmanuele, A dual characterization of Banach spaces not containing $\ell^{1}$, Bull. Polish Acad. Sci. Math. 34 (1986), no. 3–4, 155–160.
• [7] G. Emmanuele, Banach spaces in which Dunford–Pettis sets are relatively compact, Arch. Math. (Basel) 58 (1992), no. 5, 477–485.
• [8] G. Emmanuele, On Banach spaces with the Gelfand–Phillips property, III, J. Math. Pures Appl. (9) 72 (1993), no. 3, 327–333.
• [9] I. Ghenciu, Property $(wL)$ and the reciprocal Dunford–Pettis property in projective tensor products, Comment. Math. Univ. Carolin., 56 (2015), no. 3, 319–411.
• [10] I. Ghenciu and P. Lewis, The Dunford–Pettis Property, the Gelfand–Phillips Property, and $L$-sets, Colloq. Math., 106 (2006), no. 2, 311–324.
• [11] F. Mayoral, Compact sets of compact operators in the absence of $\ell_{1}$, Proc. Amer. Math. Soc. 129 (2001), no. 1, 79–82.
• [12] T. W. Palmer, Totally bounded sets of precompact linear operators, Proc. Amer. Math. Soc. 20 (1969), 101–106.
• [13] A. Pełczyński, Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 641–648.
• [14] H. Rosenthal, Point-wise compact subsets of the first Baire class, Amer. J. Math. 99 (1977), no. 2, 362–378.
• [15] W. Ruess, Compactness and collective compactness in spaces of compact operators, J. Math. Anal. Appl. 84 (1981), no. 2, 400–417.
• [16] R. Ryan, The Dunford–Pettis property and projective tensor products, Bull. Polish Acad. Sci. Math. 35 (1987), no. 11–12, 785–792.
• [17] M. Salimi and M. Moshtaghioun, The Gelfand–Phillips property in closed subspaces of some operator spaces, Banach J. Math. Anal. 5 (2011), no. 2, 84–92.
• [18] T. Schlumprecht, Limited sets in Banach spaces, Ph.D. dissertation, Ludwig-Maximilians University, Munich, 1987.