可补算子与并行和的范数不等式
Norm Inequalities for Complementable Operators and Parallel Sums
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中文总结 AI 辅助
研究希尔伯特空间可补算子的结构和定量行为，建立其范数不等式等框架，明确相关条件，探索稳定性配置，还应用于并行和设置，推导新范数分解恒等式、算子界及代数限制。
中文摘要 AI 辅助
本文研究希尔伯特空间上可补算子的结构和定量行为，重点关注其范数特征和几何轮廓。我们建立了有界线性算子与其广义舒尔补（双边缩短算子）之间范数不等式和下界关系的综合框架。在明确的算子分解和值域包含标准下，定义了有界线性算子相对于其舒尔补收缩或扩张向量的精确条件。此外，探索了$(M,N,\lambda)$-可补算子的下界性和稳定性配置，证明了下方有界的舒尔补是将单射性和下界稳定性传播到全局算子的充分条件。这些结构结果随后应用于两个有界线性算子并行和的网络理论设置。在一些特定正交条件下，我们推导了一个新的范数分解恒等式、精确的双边全局算子界以及通过摩尔 - 彭罗斯逆对道格拉斯约化解的代数限制。
英文摘要
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.
