Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T10:47:39.420Z Has data issue: false hasContentIssue false
  • English
  • Français

1-Complemented Subspaces of Spaces With 1-Unconditional Bases

Published online by Cambridge University Press:  20 November 2018

Beata Randrianantoanina
Beata Randrianantoanina*
Affiliation:
Department of Mathematics 123 Bachelor Hall Miami University Oxford, OH 45056-1641, USA
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.

We prove that if X is a complex strictly monotone sequence space with 1-unconditional basis, Y ⊆ X has no bands isometric to ℓ22 and Y is the range of norm-one projection from X, then Y is a closed linear span a family of mutually disjoint vectors in X.

We completely characterize 1-complemented subspaces and norm-one projections in complex spaces ℓp(ℓq) for 1 ≤ p,q > ∞.

Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are 1-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space X is not isomorphic to ℓp for some 1 ≤ p,q > ∞ then the only subspaces of X which are 1-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.

Keywords

46B2046B4541A65
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 6 , 01 December 1997 , pp. 1242 - 1264
Copyright
Copyright © Canadian Mathematical Society 1997

References

[ACL] Altshuler, Z., Casazza, P.G., and Lin, B.L., On symmetric basic sequences in Lorentz sequence spaces. Israel J. Math. 15(1973), 140–155.Google Scholar
[An] Ando, T., Contractive projections in Lp -spaces. Pacific J. Math. 17(1966), 391–405.Google Scholar
[B] Baronti, M., Norm–one projections onto subspaces of ℓ∞. Arch.Math. 51(1988), 242–246.Google Scholar
[BP] Baronti, M. and Papini, P.L.,Norm-one projections onto subspaces of ℓp . Ann.Mat. Pura Appl. 152(1988), 53–61.Google Scholar
[BCh] Blatter, J. and Cheney, E.W., Minimal projections on hyperplanes in sequence spaces. Ann. Mat. Pura Appl. 101(1974), 215–227.Google Scholar
[BFL] Benyamini, Y., Flinn, P., and Lewis, D.R., A spacewithout 1-unconditional basis which is 1-complemented in a space with a 1-unconditional basis. Texas Functional Analysis Seminar, 1983–84. Longhorn Notes, University of Texas Press, 1984. 145–149.Google Scholar
[C] Calvert, B., Convergence sets in reflexive Banach spaces. Proc. Amer. Math. Soc. 47(1975), 423–428.Google Scholar
[CF] Calvert, B. and Fitzpatrick, S., Characterizing ℓp and c 0 by projections onto hyperplanes. Boll. Un. Math. Ital. C (6) 5(1986), 405–410.Google Scholar
[ChP] Cheney, E.W. and Price, K.H., Minimal projections. In: Approximation Theory (Ed. Talbot, A.), Proc. Sympos., Lancaster, 1969. Academic Press, 1970. 261 – 289.Google Scholar
[D] Doust, I., Contractive projections on Lebesgue–Bochner spaces. In: Function spaces (Edwardsville, IL, 1994.; Lecture Notes in Pure and Appl. Math. 172(1995), Dekker, New York, 101–109.Google Scholar
[F] Flinn, P., On a theorem of N.J. Kalton and G.V. Wood concerning 1-complemented subspaces of spaces having an orthonormal basis. Texas Functional Analysis Seminar, 1983–84. Longhorn Notes, University of Texas Press, 1984. 135–144.Google Scholar
[GH] GrząŚlewicz, R. and Hudzik, H., Smooth points of Orlicz spaces equipped with Luxemburg norm. Math. Nachr. 155(1992), 31–45.Google Scholar
[KW] Kalton, N.J. and Wood, G.V., Orthonormal systems in Banach spaces and their applications. Math. Proc. Cambridge Philos. Soc. 79(1976), 493–510.Google Scholar
[Ko] Koldobskiî, A.L., Isometries of Lp(X; Lq ) and equimeasurability. Indiana Univ. Math. J. 40(1991), 677–705.Google Scholar
[Le] Lewis, D.R., Ellipsoids defined by Banach ideal norms. Matematika 26(1979), 18–29.Google Scholar
[Lg] Lindberg, K.J., On subspaces of Orlicz sequence spaces. Studia Math. 45(1973), 119–146.Google Scholar
[L] Lindenstrauss, J., A remark on symmetric bases. Israel J. Math. 13(1972), 317–320.Google Scholar
[LT] Lindenstrauss, J. and Tzafriri, L.,ClassicalBanach spaces I: Sequence spaces. Ergeb.Math.Grenzgeb. 92, Springer–Verlag, New York, 1978.Google Scholar
[LV] Lyubich, Y.I. and Vaserstein, L.N., Isometric embeddings between classical Banach spaces, cubature formulas, and spherical designs. Geom. Dedicata 47(1993), 327–362.Google Scholar
[R1] Rosenthal, H.P., Functional Hilbertian sums. Pacific J. Math. 124(1986), 417–467.Google Scholar
[R2] Rosenthal, H.P., On one-complemented subspaces of complex Banach spaces with one-unconditional basis, according to Kalton and Wood. Israel Seminar on Geometrical Aspects of Functional Analysis (9), 1983–84. Tel Aviv University, 1984.Google Scholar
[ST] Schneider, H. and Turner, R.E.L., Matrices Hermitian for an absolute norm. Linear and Multilinear Algebra 1(1973), 9–31.Google Scholar
[Z] Zaidenberg, M.G., Groups of isometries of Orlicz spaces. Soviet Math. Dokl. 17(1976), 432–436.Google Scholar