The determination of minimal projections and extensions in 𝐿¹

Chalmers, B. L.; Metcalf, F. T.

doi:10.1090/S0002-9947-1992-1034660-1

The determination of minimal projections and extensions in $L^ 1$
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by B. L. Chalmers and F. T. Metcalf PDF

Trans. Amer. Math. Soc. 329 (1992), 289-305 Request permission

Abstract:

Equations are derived which are shown to be necessary and sufficient for finite rank projections in ${L^1}$ to be minimal. More generally, these equations are also necessary and sufficient to determine operators of minimal norm which extend a fixed linear action on a given finite-dimensional subspace of ${L^1}$ and thus may be viewed as an extension of the Hahn-Banach theorem to higher dimensions in the ${L^1}$ setting. These equations are solved in terms of an ${L^1}$ best approximation problem and the required orthogonality conditions. Moreover, this solution has a simple geometric interpretation. Questions of uniqueness are considered and a number of examples are given to illustrate the usefulness of these equations in determining minimal projections and extensions, including the minimal ${L^1}$ projection onto the quadratics.

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