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

• Autorzy: B. L. Chalmers , F. T. Metcalf
• Języki publikacji: EN

Abstrakty

EN

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 $dU_i = -2u_i dν$.

Słowa kluczowe

EN

maximal overspace; two-dimensional spaces

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 46B04: Isometric theory of Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1991-1992
• Tom: 56
• Numer: 3
• Strony: 303-309

Daty

wydano

1992

otrzymano

1991-02-15

poprawiono

1991-07-29

Twórcy

autor

B. L. Chalmers

autor

F. T. Metcalf

• Department of Mathematics, University of California, Riverside Riverside, California 92521, U.S.A.

Bibliografia

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