Some Geometric Aspects of Operators Acting from L 1

Some Geometric Aspects of Operators Acting from L 1 We study some properties of the space [InlineMediaObject not available: see fulltext.] (L 1,X) of all continuous linear operators acting from L 1 to a Banach space X. It is proved that every operator T ∈ [InlineMediaObject not available: see fulltext.](L 1, X) ``almost'' attains its norm at the entire positive cone of functions supported at some suitable measurable subset [InlineMediaObject not available: see fulltext.], µ(A) > 0. Using this fact and a new elementary technique we prove that every operator T∈ [InlineMediaObject not available: see fulltext.] (L 1) = [InlineMediaObject not available: see fulltext.] (L 1, L 1) is uniquely represented in the form T= R+S, R, S∈ [InlineMediaObject not available: see fulltext.](L 1) , where R is representable and S possess a special property (*). Moreover, this representation generates a decomposition of the space [InlineMediaObject not available: see fulltext.] (L 1) into complemented subspaces by means of contractive projections (the fact that the subspace of all representable operators is complemented in [InlineMediaObject not available: see fulltext.] (L 1) was proved before by Z. Liu). Positivity Springer Journals

Some Geometric Aspects of Operators Acting from L 1

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Copyright © 2006 by Birkhäuser Verlag, Basel
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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