Oppenheim--Schur inequalities for causal products

TL;DR

This paper extends classical Schur and Oppenheim inequalities to causal convolutional matrix products, revealing structural parallels and introducing a unified framework for positivity-preserving operations in matrices.

Contribution

It introduces a new class of inequalities for causal matrix products, generalizing classical results and highlighting structural similarities between entrywise and convolution-based operations.

Findings

Established Oppenheim--Schur-type inequalities for causal Jury products.

Unified classical and convolutional inequalities for positive semidefinite matrices.

Provided a framework connecting classical matrix analysis with causal operator structures.

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.