2025/10/20 by Luan Yuxi, Yuxi, Luan, Rana Mondal +1
Mathematics · #46L08 #47A10 #47A63 #47B65 #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Inequalities and Applications #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2511.11588
openalex publication_date 2025/10/20 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28
We propose two interrelated advances in the theory of adjointable operators on Hilbert C*-modules. First, we give a set of equivalent, verifiable conditions characterizing positivity of general n× n block operator matrices acting on finite direct sums of Hilbert C*-modules. Our conditions generalize and remove several classical range-closedness and Moore-Penrose assumptions by expressing positivity in terms of a finite family of mixed inner-product inequalities and an explicit Gram-type factorization. Second, we investigate a parametric family of mixed-Schwarz inequalities for adjointable operators and determine optimal factor functions and constants which make these inequalities sharp; we characterize the extremal operators attaining equality in key cases. The two developments are tied together: the optimal mixed-Schwarz bounds are used to obtain sharp, computable tests in the n× n positivity criterion, and conversely the block-factorizations yield structural information used in the extremal analysis. We include applications to solvability of operator equations without Moore-Penrose inverses and spectral gap estimates for block operator generators.