A General Framework for Operator Continuity between Riesz Spaces
Abstract
We introduce and study ℱ𝒢-order continuous operators between Riesz spaces, based on Tantrawan's generalized (unbounded) order convergence. This framework includes several familiar classes of operators and allows their properties to be studied in a unified setting. We establish order boundedness and lattice-theoretic results; in particular, we show that strongly order continuous operators are automatically order bounded and prove that the modulus of an order bounded ℱ𝒢-order continuous operator remains ℱ𝒢-order continuous under suitable assumptions. The latter result gives an affirmative answer to a previously posed problem for strongly order continuous operators. We also establish a general result ensuring classical order continuity, from which order continuity results for several familiar classes of operators follow. Furthermore, we characterize, under suitable hypotheses, when the space of order bounded ℱ𝒢-order continuous operators is a band. Finally, we prove an extension theorem for positive ℱ𝒢-order continuous operators, providing a counterpart of Veksler's extension theorem in this framework.
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BibTeX
@misc{uyar2026general,
title = {{A General Framework for Operator Continuity between Riesz Spaces}},
author = {Ayse Uyar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23893},
eprint = {2609.23893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23893},
}
RIS
TY - UNPB
TI - A General Framework for Operator Continuity between Riesz Spaces
AU - Ayse Uyar
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23893
UR - https://arxiv.org/abs/2609.23893
AB - We introduce and study ℱ𝒢-order continuous operators between Riesz spaces, based on Tantrawan's generalized (unbounded) order convergence. This framework includes several familiar classes of operators and allows their properties to be studied in a unified setting. We establish order boundedness and lattice-theoretic results; in particular, we show that strongly order continuous operators are automatically order bounded and prove that the modulus of an order bounded ℱ𝒢-order continuous operator remains ℱ𝒢-order continuous under suitable assumptions. The latter result gives an affirmative answer to a previously posed problem for strongly order continuous operators. We also establish a general result ensuring classical order continuity, from which order continuity results for several familiar classes of operators follow. Furthermore, we characterize, under suitable hypotheses, when the space of order bounded ℱ𝒢-order continuous operators is a band. Finally, we prove an extension theorem for positive ℱ𝒢-order continuous operators, providing a counterpart of Veksler's extension theorem in this framework.
ER -