Geodesic
UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND l∞
Glasgow mathematical journal, Tome 53 (2011) no. 3, pp. 583-598

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.1017/S0017089511000152
Mots-clés : 46B05, 46B28, 46B25 
GHENCIU, IOANA; LEWIS, PAUL. UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND l∞. Glasgow mathematical journal, Tome 53 (2011) no. 3, pp. 583-598. doi: 10.1017/S0017089511000152
@article{10_1017_S0017089511000152,
     author = {GHENCIU, IOANA and LEWIS, PAUL},
     title = {UNCONDITIONAL {CONVERGENCE} {IN} {THE} {STRONG} {OPERATOR} {TOPOLOGY} {AND} l\ensuremath{\infty}},
     journal = {Glasgow mathematical journal},
     pages = {583--598},
     year = {2011},
     volume = {53},
     number = {3},
     doi = {10.1017/S0017089511000152},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089511000152/}
}
TY  - JOUR
AU  - GHENCIU, IOANA
AU  - LEWIS, PAUL
TI  - UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND l∞
JO  - Glasgow mathematical journal
PY  - 2011
SP  - 583
EP  - 598
VL  - 53
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089511000152/
DO  - 10.1017/S0017089511000152
ID  - 10_1017_S0017089511000152
ER  - 
%0 Journal Article
%A GHENCIU, IOANA
%A LEWIS, PAUL
%T UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND l∞
%J Glasgow mathematical journal
%D 2011
%P 583-598
%V 53
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089511000152/
%R 10.1017/S0017089511000152
%F 10_1017_S0017089511000152

[1] 1.Banach, S., Theorie des operations linéaires (Monografie Matematyczme, Warsaw, 1932). Google Scholar

[2] 2.Bator, E. and Lewis, P., Complemented spaces of operators, Bull. Pol. Acad. Sci. Math. 50 (4) (2002), 413–416. Google Scholar

[3] 3.Bator, E. M., Lewis, P. W. and Slavens, D. R., Vector measures, c , and (sb) operators, Bull. Pol. Acad. Sci. Math. 54 (1) (2006), 63–73. Google Scholar

[4] 4.Bessaga, C. and Pelczynski, A., On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–174. Google Scholar | DOI

[5] 5.Diestel, J., Sequences and series in Banach spaces, Graduate Texts in Mathematics, no. 92 (Springer-Verlag, Berlin, 1984). Google Scholar

[6] 6.Diestel, J. and Uhl, J. J. Jr., Vector measures, Mathematical Surveys 15 (American Mathematical Society, Providence, RI, 1977). Google Scholar | DOI

[7] 7.Drewnowski, L., Copies of ℓ in an operator space, Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526. Google Scholar

[8] 8.Emmanuele, G., Remarks on the uncomplemented subspace W(E, F), J. Funct. Anal. 99 (1991), 125–130. Google Scholar

[9] 9.Emmanuele, G., A remark on the containment of c in spaces of compact operators, Math. Proc. Camb. Phil. Soc. 111 (1992), 331–335. Google Scholar

[10] 10.Emmanuele, G., About the position of K(X*, Y) inside L(X*, Y), Atti. Sem. Mat. Fis. Univ. Modena 42 (1) (1994), 123–133. Google Scholar

[11] 11.Feder, M., On the non-existence of a projection onto the space of compact operators, Canad. Math. Bull. 25 (1982), 78–81. Google Scholar | DOI

[12] 12.Ghenciu, I., Complemented spaces of operators, Proc. Amer. Math. Soc. 133 (9) (2005), 2621–2623. Google Scholar | DOI

[13] 13.Ghenciu, I. and Lewis, P., The embeddability of c in spaces of operators, Bull. Pol. Acad. Sci. Math. 56 (3–4) (2008), 239–256. Google Scholar | DOI

[14] 14.Ghenciu, I. and Lewis, P., Dunford-Pettis properties and spaces of operators, Canad. Math. Bull. 52 (2009), 213–223. Google Scholar | DOI

[15] 15.John, K., On the uncomplemented subspace K(X, Y), Czech. Math. J. 42 (1992), 167–173. Google Scholar | DOI

[16] 16.Kalton, N., Spaces of compact operators, Math. Ann. 208 (1974), 267–278. Google Scholar

[17] 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] 18.Lewis, P., Spaces of operators and c , Studia Math. 145 (2001), 213–218. Google Scholar

[19] 19.Lewis, P. and Schulle, P., Non-complemented spaces of linear operators, vector measures, and c , Canad. Math. Bull., to appear. Google Scholar

[20] 20.Pietsch, A., Nuclear locally convex spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 66 (Springer-Verlag, New York, 1972). Google Scholar

[21] 21.Pitt, H. R., A note on bilinear forms, J. Lond. Math. Soc. 11 (1936), 174–180. Google Scholar | DOI

[22] 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] 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] 24.Singer, I., Bases in Banach spaces (Springer, New Mexico, 1981). Google Scholar

Cité par Sources :