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Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we give conditions ensuring that a semigroup consisting of kernel operators has a Laplace transform which also consists of kernel operators....
Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we...
A weak form of the Henstock Lemma for the -integrable functions is given. This allows to prove the existence of a scalar Volterra derivative for the -integral. Also the -integrable functions are characterized by means of Pettis integrability and a condition involving finite pseudopartitions.
An abstract characterization of Orlicz-Kantorovich lattices constructed by a measure with values in the ring of measurable functions is presented.
An integral for vector-valued functions on a σ-finite outer regular quasi-Radon measure space is defined by means of partitions of unity and it is shown that it is equivalent to the McShane integral. The multipliers for both the McShane and Pettis integrals are characterized.
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