Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-05T10:14:39.928Z Has data issue: false hasContentIssue false
  • English
  • Français

UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND ℓ∞

Published online by Cambridge University Press:  10 March 2011

IOANA GHENCIU  and
PAUL LEWIS
IOANA GHENCIU
Affiliation:
University of Wisconsin - River Falls, Department of Mathematics, River Falls, WI 54022-5001 e-mail: [email protected]
PAUL LEWIS
Affiliation:
University of North Texas, Department of Mathematics, Box 311430, Denton, Texas 76203-1430 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study non-complemented spaces of operators and the embeddability of ℓ∞ in the spaces of operators L(X, Y), K(X, Y) and Kw*(X*, Y). Results of Bator and Lewis [2, 3] (Bull. Pol. Acad. Sci. Math.50(4) (2002), 413–416; Bull. Pol. Acad. Sci. Math.549(1) (2006), 63–73), Emmanuele [8–10] (J. Funct. Anal.99 (1991), 125–130; Math. Proc. Camb. Phil. Soc.111 (1992), 331–335; Atti. Sem. Mat. Fis. Univ. Modena42(1) (1994), 123–133), Feder [11] (Canad. Math. Bull.25 (1982), 78–81) and Kalton [16] (Math. Ann.208 (1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spaces W(X, Y) and K(X, Y) in the space L(X, Y), as well as the complementation of K(X, Y) in W(X, Y). A fundamental result of Drewnowski [7] (Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach space version of the Phillips Lemma.

Keywords

46B0546B2846B25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 583 - 598
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Banach, S., Theorie des operations linéaires (Monografie Matematyczme, Warsaw, 1932).Google Scholar
2.Bator, E. and Lewis, P., Complemented spaces of operators, Bull. Pol. Acad. Sci. Math. 50 (4) (2002), 413–416.Google Scholar
3.Bator, E. M., Lewis, P. W. and Slavens, D. R., Vector measures, c 0, and (sb) operators, Bull. Pol. Acad. Sci. Math. 54 (1) (2006), 63–73.Google Scholar
4.Bessaga, C. and Pelczynski, A., On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–174.CrossRefGoogle Scholar
5.Diestel, J., Sequences and series in Banach spaces, Graduate Texts in Mathematics, no. 92 (Springer-Verlag, Berlin, 1984).Google Scholar
6.Diestel, J. and Uhl, J. J. Jr., Vector measures, Mathematical Surveys 15 (American Mathematical Society, Providence, RI, 1977).CrossRefGoogle Scholar
7.Drewnowski, L., Copies of ℓ∞ in an operator space, Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526.Google Scholar
8.Emmanuele, G., Remarks on the uncomplemented subspace W(E, F), J. Funct. Anal. 99 (1991), 125–130.Google Scholar
9.Emmanuele, G., A remark on the containment of c 0 in spaces of compact operators, Math. Proc. Camb. Phil. Soc. 111 (1992), 331–335.Google Scholar
10.Emmanuele, G., About the position of Kw*(X*, Y) inside Lw*(X*, Y), Atti. Sem. Mat. Fis. Univ. Modena 42 (1) (1994), 123–133.Google Scholar
11.Feder, M., On the non-existence of a projection onto the space of compact operators, Canad. Math. Bull. 25 (1982), 78–81.CrossRefGoogle Scholar
12.Ghenciu, I., Complemented spaces of operators, Proc. Amer. Math. Soc. 133 (9) (2005), 2621–2623.CrossRefGoogle Scholar
13.Ghenciu, I. and Lewis, P., The embeddability of c 0 in spaces of operators, Bull. Pol. Acad. Sci. Math. 56 (3–4) (2008), 239–256.CrossRefGoogle Scholar
14.Ghenciu, I. and Lewis, P., Dunford-Pettis properties and spaces of operators, Canad. Math. Bull. 52 (2009), 213–223.CrossRefGoogle Scholar
15.John, K., On the uncomplemented subspace K(X, Y), Czech. Math. J. 42 (1992), 167–173.CrossRefGoogle Scholar
16.Kalton, N., Spaces of compact operators, Math. Ann. 208 (1974), 267–278.Google Scholar
17.Kupka, J., A short proof and generalization of a measure theoretic disjointization lemma, Proc. Amer. Math. Soc. 45 (1) (1974), 70–72.Google Scholar
18.Lewis, P., Spaces of operators and c 0, Studia Math. 145 (2001), 213–218.Google Scholar
19.Lewis, P. and Schulle, P., Non-complemented spaces of linear operators, vector measures, and c 0, Canad. Math. Bull., to appear.Google Scholar
20.Pietsch, A., Nuclear locally convex spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 66 (Springer-Verlag, New York, 1972).Google Scholar
21.Pitt, H. R., A note on bilinear forms, J. Lond. Math. Soc. 11 (1936), 174–180.CrossRefGoogle Scholar
22.Rosenthal, H., On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36.Google Scholar
23.Ruess, W., Duality and geometry of spaces of compact operators, functional analysis: Surveys and recent results III. in Proceedings of 3rd Paderborn Conference 1983, North-Holland Mathematics Studies no. 90 (North-Holland Publ. Co., New York, NY, 1984), 59–78.Google Scholar
24.Singer, I., Bases in Banach spaces (Springer, New Mexico, 1981).Google Scholar