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Block Positivity and Optimal Mixed-Schwarz Inequalities on Hilbert $C^*$-Modules

Luan Yuxi, Rana Mondal

TL;DR

The paper delivers two intertwined advances for adjointable operators on Hilbert C*-modules: (i) a practical, verifiable criterion for positivity of arbitrary n-by-n block operator matrices that avoids closed-range or Moore-Penrose assumptions by using mixed inner-product inequalities and a Gram-type factorization; and (ii) an optimization framework for a parametric mixed-Schwarz inequality, identifying sharp constants in the approximable/normal cases and enabling explicit extremal characterizations. The two threads are linked: sharp mixed-Schwarz bounds feed into computable positivity tests, while block-factorizations inform the extremal analysis. Concrete applications include Moore-Penrose-free solvability criteria for operator equations and explicit coercivity/spectral-gap estimates for block generators, supplemented by algorithmic procedures and numerical illustrations. Together, these results extend classical operator theory to Hilbert C*-modules, providing both structural insight and computational tools for noncommutative analysis and PDE-related problems.

Abstract

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.

Block Positivity and Optimal Mixed-Schwarz Inequalities on Hilbert $C^*$-Modules

TL;DR

The paper delivers two intertwined advances for adjointable operators on Hilbert C*-modules: (i) a practical, verifiable criterion for positivity of arbitrary n-by-n block operator matrices that avoids closed-range or Moore-Penrose assumptions by using mixed inner-product inequalities and a Gram-type factorization; and (ii) an optimization framework for a parametric mixed-Schwarz inequality, identifying sharp constants in the approximable/normal cases and enabling explicit extremal characterizations. The two threads are linked: sharp mixed-Schwarz bounds feed into computable positivity tests, while block-factorizations inform the extremal analysis. Concrete applications include Moore-Penrose-free solvability criteria for operator equations and explicit coercivity/spectral-gap estimates for block generators, supplemented by algorithmic procedures and numerical illustrations. Together, these results extend classical operator theory to Hilbert C*-modules, providing both structural insight and computational tools for noncommutative analysis and PDE-related problems.

Abstract

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 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 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.

Paper Structure

This paper contains 8 sections, 13 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Foundations and Notation
  3. Main results and technical developments
  4. Examples, Numerical Illustrations, and Algorithmic Procedures
  5. Comparison with classical $2\times2$ reductions.
  6. Applications: Solvability of Operator Equations and Spectral--Gap Estimates
  7. Conclusion
  8. Appendix: Technical Lemmas and Proofs

Key Result

Theorem 3.1

Let $E_1,\dots,E_n$ be Hilbert C*-modules over the same unital C*-algebra $\mathscr A$. Let be a block operator matrix with $T_{ij}\in\mathcal{L}(E_j,E_i)$ and suppose that $T_{ii}\ge0$ for each $i$. Define the scalar quadratic forms Assume that for all $i\ne j$ and all $x_i\in E_i,\ x_j\in E_j$ the cross--entry seminorms satisfy Then $T\ge0$. Conversely, if $T\ge0$ then ineq:cross holds for ev

Theorems & Definitions (35)

  • Theorem 3.1: $n\times n$ positivity via cross--entry seminorms
  • Remark 3.2
  • Lemma 3.3: Schur--chain lemma
  • proof
  • Lemma 3.4: Gram--factor construction
  • proof
  • Remark 3.5
  • proof : Proof of Theorem \ref{['thm:main-nxn']}
  • Proposition 3.6
  • proof
  • ...and 25 more