Block Positivity and Optimal Mixed-Schwarz Inequalities on Hilbert $C^*$-Modules
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arXiv
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| Natura: | Preprint |
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2025
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| author | Yuxi, Luan Mondal, Rana |
| author_facet | Yuxi, Luan Mondal, Rana |
| contents | 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 $n\times n$ 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 $n\times n$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Block Positivity and Optimal Mixed-Schwarz Inequalities on Hilbert $C^*$-Modules Yuxi, Luan Mondal, Rana Functional Analysis Operator Algebras 46L08, 47A63, 47B65, 47A10 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 $n\times n$ 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 $n\times n$ 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. |
| title | Block Positivity and Optimal Mixed-Schwarz Inequalities on Hilbert $C^*$-Modules |
| topic | Functional Analysis Operator Algebras 46L08, 47A63, 47B65, 47A10 |
| url | https://arxiv.org/abs/2511.11588 |