An Optimal Domain Theorem for Comonotonically Additive Vector-Valued Operators
- 1Universidad de Guadalajara, Centro Universitario de Ciencias Exactas e Ingenierías, Departamento de Matemáticas, Guadalajara 44430, Jalisco, Mexico
- 2Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain
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.
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BibTeX
@article{ascencio2026optimal,
title = {{An Optimal Domain Theorem for Comonotonically Additive Vector-Valued Operators}},
author = {Luis Gustavo Alvarez Ascencio and Celia Avalos-Ramos and Enrique Alfonso Sánchez Pérez},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203790},
url = {https://www.mdpi.com/2227-7390/14/20/3790},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - An Optimal Domain Theorem for Comonotonically Additive Vector-Valued Operators
AU - Luis Gustavo Alvarez Ascencio
AU - Celia Avalos-Ramos
AU - Enrique Alfonso Sánchez Pérez
PY - 2026
DA - 2026/10/19
JO - Mathematics
DO - 10.3390/math14203790
UR - https://www.mdpi.com/2227-7390/14/20/3790
AB - 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.
SN - 2227-7390
ER -