Summary
Under study is the existence of averaging operators determined by measurable maps φ from a measure space (S, Σ, μ) into an arbitrary Hausdorff topological space T. The map φ induces a continuous map φe from the space Cb(T) into the normed (Banach) function space Lϱ = Lϱ(S, Σ, μ) defined by φe(f)=foφ for all f ε Cb(T). An integral representation for such operators is first studied. The existence is then determined by the existence of an averaging operator U1 for the restriction of φ to a certain measurable subset B1 of S. Utilizing a representation of Lϱ(S, Σ, μ) as a Banach function space over a compact extremally disconnected Hausdorff space Ŝ, we are able to give a definition for the concept of plural points and irreducible map. A significant upper bound is given for the operator U1. Finally conditions are considered under which no bounded projection from Lϱ onto the range of φe may exist. From a topological point of view the development is pursued in a general setting. Averaging operators have recently been used for the study of injective Banach spaces of the type Cb(T) and in non-linear prediction and approximation theory relative to Tshebyshev subspaces of Lϱ.
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Entrata in Redazione l’ll settembre 1975.
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Alò, R.A., de Korvin, A. & Roberts, C. Averaging operators on normed köthe spaces. Annali di Matematica 112, 33–48 (1977). https://doi.org/10.1007/BF02413474
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DOI: https://doi.org/10.1007/BF02413474