Let P:X → V be a projection from a real Banach space X onto a subspace V and let S ⊂ X. In this setting, one can ask if S is left invariant under P, i.e., if PS ⊂ S. If V is finite-dimensional and S is a cone with particular structure, then the occurrence of the imbedding PS ⊂ S can be characterized through a geometric description. This characterization relies heavily on the structure of S, or, more specifically, on the structure of the cone S* dual to S. In this paper, we remove the structural assumptions on S* and characterize the cases where PS ⊂ S. We note that the (so-called) q-monotone shape forms a cone that (lacks structure and thus) serves as an application for our characterization.
Existence Result Positive Operator Banach Lattice Real Banach Space Subspace Versus
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B. Chalmers, D. Mupasiri, and M. P. Prophet, “A characterization and equations for minimal shape-preserving projections,” J. Approxim. Theory, 138, 184–196 (2006).MathSciNetzbMATHCrossRefGoogle Scholar