The Mackey topology and complemented subspaces of Lorentz sequence spaces 𝑑(𝑤,𝑝) for 0<𝑝<1

Nawrocki, M.; Ortyński, A.

doi:10.1090/S0002-9947-1985-0768736-7

The Mackey topology and complemented subspaces of Lorentz sequence spaces $d(w,p)$ for $0<p<1$
HTML articles powered by AMS MathViewer

by M. Nawrocki and A. Ortyński PDF

Trans. Amer. Math. Soc. 287 (1985), 713-722 Request permission

Abstract:

In this paper we continue the study of Lorentz sequence spaces $d(w,p)$, $0 < p < 1$, initiated by N. Popa [8]. First we show that the Mackey completion of $d(w,p)$ is equal to $d(v,1)$ for some sequence $v$. Next, we prove that if $d(w,p) \not \subset {l_1}$, then it contains a complemented subspace isomorphic to ${l_p}$. Finally we show that if $\lim {n^{ - 1}}\left (\sum \nolimits _{i = 1}^n {w_i}\right )^{1/p} = \infty$, then every complemented subspace of $d(w,p)$ with symmetric bases is isomorphic to $d(w,p)$.

References

Zvi Altshuler, P. G. Casazza, and Bor Luh Lin, On symmetric basic sequences in Lorentz sequence spaces, Israel J. Math. 15 (1973), 140–155. MR 328553, DOI 10.1007/BF02764600
P. G. Casazza and Bor Luh Lin, On symmetric basic sequences in Lorentz sequence spaces. II, Israel J. Math. 17 (1974), 191–218. MR 348443, DOI 10.1007/BF02882238
P. G. Casazza and Bor Luh Lin, Projections on Banach spaces with symmetric bases, Studia Math. 52 (1974), 189–193. MR 350384, DOI 10.4064/sm-52-2-189-193
D. J. H. Garling, On symmetric sequence spaces, Proc. London Math. Soc. (3) 16 (1966), 85–106. MR 192311, DOI 10.1112/plms/s3-16.1.85
H. Elton Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. MR 0493279
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056

Projections in locally bounded spaces with symmetric bases

Nicolae Popa, Basic sequences and subspaces in Lorentz sequence spaces without local convexity, Trans. Amer. Math. Soc. 263 (1981), no. 2, 431–456. MR 594418, DOI 10.1090/S0002-9947-1981-0594418-7
Stefan Rolewicz, Metric linear spaces, 2nd ed., PWN—Polish Scientific Publishers, Warsaw; D. Reidel Publishing Co., Dordrecht, 1984. MR 802450
Joel H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces, Duke Math. J. 43 (1976), no. 1, 187–202. MR 500100