On Parallel Sums of Set-Valued Operators: Monotonicity, Paramonotonicity, and Decomposition Invariance
Abstract
In this paper, we investigate structural properties of the parallel sum of set-valued operators in a real Hilbert space, with particular emphasis on monotonicity, maximal monotonicity, and paramonotonicity. We examine the decomposition-invariance property of the parallel sum of two paramonotone operators and provide a direct proof based on monotonicity and the equality of admissible decompositions. This approach establishes the required orthogonality relations without invoking the Attouch--Th{é}ra Duality duality framework. We also revisit the classical maximal-monotonicity property of the parallel sum and present an alternative graph-based derivation involving an auxiliary shifted operator. Furthermore, we study the parallel sum of a set-valued operator and the normal cone operator associated with a closed linear subspace. Explicit characterizations of its values, domain, and range are presented, and a decomposition-independent representation is obtained under paramonotonicity through the orthogonality of the underlying subspaces. These results provide a unified and self-contained treatment of selected properties of parallel sums and clarify the role of paramonotonicity in their structural behavior.
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BibTeX
@misc{alwadani2026parallel,
title = {{On Parallel Sums of Set-Valued Operators: Monotonicity, Paramonotonicity, and Decomposition Invariance}},
author = {Salihah Thabet Alwadani},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05523},
eprint = {2611.05523},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05523},
}
RIS
TY - UNPB
TI - On Parallel Sums of Set-Valued Operators: Monotonicity, Paramonotonicity, and Decomposition Invariance
AU - Salihah Thabet Alwadani
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05523
UR - https://arxiv.org/abs/2611.05523
AB - In this paper, we investigate structural properties of the parallel sum of set-valued operators in a real Hilbert space, with particular emphasis on monotonicity, maximal monotonicity, and paramonotonicity. We examine the decomposition-invariance property of the parallel sum of two paramonotone operators and provide a direct proof based on monotonicity and the equality of admissible decompositions. This approach establishes the required orthogonality relations without invoking the Attouch--Th{é}ra Duality duality framework. We also revisit the classical maximal-monotonicity property of the parallel sum and present an alternative graph-based derivation involving an auxiliary shifted operator. Furthermore, we study the parallel sum of a set-valued operator and the normal cone operator associated with a closed linear subspace. Explicit characterizations of its values, domain, and range are presented, and a decomposition-independent representation is obtained under paramonotonicity through the orthogonality of the underlying subspaces. These results provide a unified and self-contained treatment of selected properties of parallel sums and clarify the role of paramonotonicity in their structural behavior.
ER -