A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces

B. L. Chalmers; F. T. Metcalf

Annales Polonici Mathematici, Tome 57 (1992), p. 303-309 / Harvested from The Polish Digital Mathematics Library

Résumé

It follows easily from a result of Lindenstrauss that, for any real twodimensional subspace v of L¹, the relative projection constant λ(v;L¹) of v equals its (absolute) projection constant $λ (v) = s u p_{X} λ (v; X)$ . The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in $L^{\infty} (ν)$ and isometric to v and a projection $P_{\infty}$ from C ⊕ V onto V such that $∥ P_{\infty} ∥ = ∥ P ₁ ∥$ , where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if $P ₁ = \sum_{i = 1}^{2} U_{i} \otimes v_{i}$ , then $P_{\infty} = \sum_{i = 1}^{2} u_{i} \otimes V_{i}$ , where $d V_{i} = 2 v_{i} d ν$ and $d U_{i} = - 2 u_{i} d ν$ .

Publié le : 1992-01-01

Zbl 0808.46016

EUDML-ID : urn:eudml:doc:262300

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     author = {B. L. Chalmers and F. T. Metcalf},
     title = {A simple formula showing L$^1$ is a maximal overspace for two-dimensional real spaces},
     journal = {Annales Polonici Mathematici},
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     year = {1992},
     pages = {303-309},
     zbl = {0808.46016},
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B. L. Chalmers; F. T. Metcalf. A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces. Annales Polonici Mathematici, Tome 57 (1992) pp. 303-309. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv56z3p303bwm/

Bibliographie

[000] [1] B. L. Chalmers, Absolute projection constant of the linear functions in a Lebesgue space, Constr. Approx. 4 (1988), 107-110. | Zbl 0647.41027

[001] [2] B. L. Chalmers and F. T. Metcalf, The determination of minimal projections and extensions in L¹, Trans. Amer. Math. Soc. 329 (1992), 289-305. | Zbl 0753.41018

[002] [3] B. L. Chalmers, F. T. Metcalf, B. Shekhtman and Y. Shekhtman, The projection constant of a two-dimensional real Banach space is no greater than 4/3, submitted. | Zbl 0970.46007

[003] [4] C. Franchetti and E. W. Cheney, Minimal projections in 𝓛₁-spaces, Duke Math. J. 43 (1976), 501-510. | Zbl 0335.41021

[004] [5] J. Lindenstrauss, On the extension of operators with a finite-dimensional range, Illinois J. Math. 8 (1964), 488-499. | Zbl 0132.09803

[005] [6] D. Yost, L₁ contains every two-dimensional normed space, Ann. Polon. Math. 49 (1988), 17-19. | Zbl 0679.46016