An Optimal Domain Theorem for Comonotonically Additive Vector-Valued Operators

Source abstract

In this paper, we consider a vector-valued operator T taking values in a quasi-Banach space X, assuming T is comonotonically additive on the positive cone. In this setting, we establish an optimal domain theorem through vector-valued Choquet integration. More precisely, given a σ-order continuous λ-quasi-Banach function space X with S(Σ)⊂X, an order continuous Banach lattice Y, and an operator T:X→Y that is comonotonically additive on the positive cone, we identify natural structural assumptions under which T can be represented as integration with respect to a vector capacity. In particular, assuming that T is Jordan decomposable, monotone, submodular, and lower semi-continuous, we construct a vector capacity ΛT such that T factors through the Choquet-type space Lqv1(ΛT) via the corresponding integration operator. Under suitable assumptions on the null sets of the underlying capacity, we prove an analogue of the classical Optimal Domain Theorem by showing that Lqv1(ΛT) is the largest σ-order continuous quasi-Banach function space to which T admits a continuous extension preserving the assumed structural properties. Consequently, Lqv1(ΛT) is the optimal domain of T, extending the classical Optimal Domain Theorem from continuous linear operators to comonotonically additive operators on the positive cone in the setting of Choquet integration.