arXiv:2611.05523v1 Announce Type: new 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{\'e}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.
Continue to arXiv Mathematics →You are being forwarded in 6s — stay on this page