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Oppenheim--Schur inequalities for causal products

Dominique Guillot, Javad Mashreghi, Prateek Kumar Vishwakarma

TL;DR

A class of Oppenheim--Schur-type inequalities for the convolutional Jury product of positive semidefinite matrices is established and unified inequalities that simultaneously recover the classical Schur and Oppenheim bounds as well as their convolutional Jury counterparts are proved.

Abstract

We establish a class of Oppenheim--Schur-type inequalities for the convolutional Jury product of positive semidefinite matrices. These results extend to a causal convolutional setting the classical Schur and Oppenheim inequalities associated with the Hadamard product. Our approach highlights structural parallels between entrywise and convolution-based matrix operations, revealing how positivity constraints interact with causality. Building on this perspective, we introduce a broader family of causal matrix products and prove unified inequalities that simultaneously recover the classical Schur and Oppenheim bounds as well as their convolutional Jury counterparts. These results provide a common framework for understanding positivity-preserving matrix products and suggest further connections between classical matrix analysis and causal operator structures.

Oppenheim--Schur inequalities for causal products

TL;DR

A class of Oppenheim--Schur-type inequalities for the convolutional Jury product of positive semidefinite matrices is established and unified inequalities that simultaneously recover the classical Schur and Oppenheim bounds as well as their convolutional Jury counterparts are proved.

Abstract

We establish a class of Oppenheim--Schur-type inequalities for the convolutional Jury product of positive semidefinite matrices. These results extend to a causal convolutional setting the classical Schur and Oppenheim inequalities associated with the Hadamard product. Our approach highlights structural parallels between entrywise and convolution-based matrix operations, revealing how positivity constraints interact with causality. Building on this perspective, we introduce a broader family of causal matrix products and prove unified inequalities that simultaneously recover the classical Schur and Oppenheim bounds as well as their convolutional Jury counterparts. These results provide a common framework for understanding positivity-preserving matrix products and suggest further connections between classical matrix analysis and causal operator structures.
Paper Structure (7 sections, 11 theorems, 46 equations)

This paper contains 7 sections, 11 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. The Jury product and inequalities
  3. Organization of the paper
  4. The special case
  5. The transformation $\widetilde{A}$
  6. The general proof
  7. The inequalities for causal products

Key Result

Theorem 2.1

For any integer $N\geq 2$, we have for all positive semidefinite $A=(a_{jk}),B=(b_{jk})\in \mathbb{C}^{N\times N}$.

Theorems & Definitions (22)

  • Theorem 2.1: The inequality
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Theorem 5.1
  • ...and 12 more