Optimal extensions for positive order continuous operators on Banach function spaces
The aim of this paper is to generalize known results on the extension of operators to spaces of integrable functions with respect to a vector measure that represent their so-called optimal domains. This procedure has been used fruitfully since the end of the 1990s, and several relevant results has been obtained since then, both from the point of view of particular applications to relevant operators, for example, the Hardy operator, also by the author, and from the point of view of the construction of an abstract setting that provides useful tools in Banach lattice theory and operator theory. The present paper provides what in a sense can be considered as the last step in this construction, pushing to the limits the requirements for this technique to work. The use of vector measures on \(\delta\)-rings is the key for this maximal extension, and their use gives clear and optimal results.NEWLINENEWLINEFrom the technical point of view, the main results of the paper (Theorem 3.1 and Theorem 4.1) can be proved because of the use of \(\delta\)-rings, in which the author is a well-known specialist. The requirements that are needed for the applications of the results that are already known make it necessary to assume stronger properties in the present paper, both on the spaces and the operators, than the ones that are otherwise needed. Concretely, order continuity of the space or the operator, operators being \(\mu\)-determined and \(\sigma\)-finiteness of the scalar measure \(\mu\) in which the Banach function space \(X(\mu)\) is based are not necessary for the results that are presented by the author. These conditions are in a sense substituted by the order properties of the space, using the information on the Fatou and related properties -- superorder density, \(\sigma\)-order w-continuity -- that is nowadays known for the spaces \(L^1_w(\nu)\) of a vector measure \(\nu\). This makes the assumption on the positivity of the involved operators necessary for the results to be true.NEWLINENEWLINEThe paper also contains some relevant results on kernel operators which are applications of the previous ones for general operators (last Section 5). The examples in this section are new and particularly interesting, and they add an extra value to the paper. Both cases that are studied in the paper (\(\sigma\)-finite and discrete measure cases) have their corresponding examples in this last section.
scientific article; zbMATH DE number 6349681
scientific article