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

B. L. Chalmers; F. T. Metcalf

Annales Polonici Mathematici (1992)

  • Volume: 56, Issue: 3, page 303-309
  • ISSN: 0066-2216

Abstract

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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 ∞ ( ν ) and isometric to v and a projection P ∞ from C ⊕ V onto V such that ∥ P ∞ ∥ = ∥ P ₁ ∥ , where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if P ₁ = ∑ i = 1 2 U i ⊗ v i , then P ∞ = ∑ i = 1 2 u i ⊗ V i , where d V i = 2 v i d ν and d U i = - 2 u i d ν .

How to cite

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B. L. Chalmers, and F. T. Metcalf. "A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces." Annales Polonici Mathematici 56.3 (1992): 303-309. <http://eudml.org/doc/262300>.

@article{B1992,
abstract = {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 $λ(v) = sup_X λ(v;X)$. The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in $L^∞(ν)$ and isometric to v and a projection $P_∞$ from C ⊕ V onto V such that $∥P_∞∥ = ∥P₁∥$, where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if $P₁ = ∑_\{i=1\}^2 U_i ⊗ v_i$, then $P_∞ = ∑_\{i=1\}^2 u_i ⊗ V_i$, where $dV_i = 2v_i dν$ and $dU_i = -2u_i dν$.},
author = {B. L. Chalmers, F. T. Metcalf},
journal = {Annales Polonici Mathematici},
keywords = {maximal overspace; two-dimensional spaces; relative projection constant},
language = {eng},
number = {3},
pages = {303-309},
title = {A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces},
url = {http://eudml.org/doc/262300},
volume = {56},
year = {1992},
}

TY - JOUR
AU - B. L. Chalmers
AU - F. T. Metcalf
TI - A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces
JO - Annales Polonici Mathematici
PY - 1992
VL - 56
IS - 3
SP - 303
EP - 309
AB - 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 $λ(v) = sup_X λ(v;X)$. The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in $L^∞(ν)$ and isometric to v and a projection $P_∞$ from C ⊕ V onto V such that $∥P_∞∥ = ∥P₁∥$, where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if $P₁ = ∑_{i=1}^2 U_i ⊗ v_i$, then $P_∞ = ∑_{i=1}^2 u_i ⊗ V_i$, where $dV_i = 2v_i dν$ and $dU_i = -2u_i dν$.
LA - eng
KW - maximal overspace; two-dimensional spaces; relative projection constant
UR - http://eudml.org/doc/262300
ER -

References

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  1. [1] B. L. Chalmers, Absolute projection constant of the linear functions in a Lebesgue space, Constr. Approx. 4 (1988), 107-110. Zbl0647.41027
  2. [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. Zbl0753.41018
  3. [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. Zbl0970.46007
  4. [4] C. Franchetti and E. W. Cheney, Minimal projections in 𝓛₁-spaces, Duke Math. J. 43 (1976), 501-510. Zbl0335.41021
  5. [5] J. Lindenstrauss, On the extension of operators with a finite-dimensional range, Illinois J. Math. 8 (1964), 488-499. Zbl0132.09803
  6. [6] D. Yost, L₁ contains every two-dimensional normed space, Ann. Polon. Math. 49 (1988), 17-19. Zbl0679.46016

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