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arXiv · 2607.10166

Norm Inequalities for Complementable Operators and Parallel Sums

Sachin Manjunath Naik·P. Sam Johnson
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Abstract

This paper investigates the structural and quantitative behaviors of complementable operators on Hilbert spaces, focusing on their norm characteristics and geometric profiles. We establish a comprehensive framework of norm inequalities and lower-bound relationships between a bounded linear operator and its generalized Schur complement (bilateral shorted operator). Under explicit operator factorization and range inclusion criteria, we define the exact conditions under which a bounded linear operator contracts or expands vectors relative to its Schur complement. Furthermore, we explore the lower boundedness and stability configurations of $(M, N, λ)$-complementable operators, proving that a bounded-below Schur complement acts as a sufficient condition to propagate injectivity and lower-bounded stability to the global operator. These structural results are subsequently applied to the network-theoretic setting of the parallel sum of two bounded linear operators. With some specific orthogonality conditions, we derive a novel norm decomposition identity, sharp two-sided global operator bounds, and algebraic restrictions on Douglas reduced solutions via Moore-Penrose inverses.

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Sachin Manjunath Naik, P. Sam Johnson. 2026-07-11. Norm Inequalities for Complementable Operators and Parallel Sums. https://arxiv.org/abs/2607.10166

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