Averaging operators on normed köthe spaces

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 φ^e from the space C_b(T) into the normed (Banach) function space L_ϱ = L_ϱ(S, Σ, μ) defined by φ^e(f)=foφ for all f ε C_b(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 L_ϱ(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 L_ϱ onto the range of φ^e 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 C_b(T) and in non-linear prediction and approximation theory relative to Tshebyshev subspaces of L_ϱ. Entrata in Redazione l’ll settembre 1975.