An integral representation of randomized probabilities and its applications
We study different types of asymptotic behaviour in the set of (infinite dimensional) nonhomogeneous chains of stochastic operators acting on L¹(μ) spaces. In order to examine its structure we consider different norm and strong operator topologies. To describe the nature of the set of nonhomogeneous chains of Markov operators with a particular limit behaviour we use the category theorem of Baire. We show that the geometric structure of the set of those stochastic operators which have asymptotically...
Asymptotic convergence theorems for nonnegative operators on Banach lattices, on , on C(X) and on are proved. The general results are applied to a class of integral operators on L¹.
New sufficient conditions for asymptotic stability of Markov operators are given. These criteria are applied to a class of Volterra type integral operators with advanced argument.
We prove that a biseparating map between spaces of vector-valued continuous functions is usually automatically continuous. However, we also discuss special cases when this is not true.
Convolutional representations of the commutant of the partial integration operators in the space of continuous functions in a rectangle are found. Necessary and sufficient conditions are obtained for two types of representing functions, to be the operators in the commutant continuous automorphisms. It is shown that these conditions are equivalent to the requirement that the considered representing functions be joint cyclic elements of the partial integration operators.