Abstract. Let K be a closed convex bounded subset of a Banach space X and let T : K → K be a continuous affine mapping. In this note, we show that (a) if T is nonexpansive then it has a fixed point, (b) if T has only one fixed point then the mapping A = (I + T
Let E be a Banach function space on a probability measure space (Ω, Σ, µ). Let X be a Banach space and E(X) be the associated Köthe-Bochner space. An operator on E(X) is called a multiplication operator if it is given by multiplication by a function in L ∞ (µ). In the main result of this paper, we show that an operator T on E(X) is a multiplication operator if and only if T commutes with L ∞ (µ) and leaves invariant the cyclic subspaces generated by the constant vector-valued functions in E(X). As a corollary we show that this is equivalent to T satisfying a functional equation considered by Calabuig, Rodríguez, Sánchez-Pérez in [3].
Let A and B be f -algebras with unit elements e A and e B respectively. A positive operator T from A to B satisfying T (e A ) = e B is called a Markov operator. In this definition we replace unit elements with weak order units and, in this case, call T to be a weak Markov operator. In this paper, we characterize extreme points of the weak Markov operators.Mathematics Subject Classification (2010) 47B38 · 46B42
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