$\hat{H}$-eigenvalues of Hermitian tensors and some applications

TL;DR

This paper introduces $\\hat{H}$-eigenvalues for Hermitian tensors, establishes criteria for positive definiteness, and applies these concepts to complex geometry, revalidating recent algebraic results.

Contribution

It defines a new eigenvalue concept for Hermitian tensors, provides checkable inclusion sets and criteria for positive definiteness, and applies these to complex differential geometry.

Findings

Established checkable inclusion sets for $\hat{H}$-eigenvalues.

Derived criteria for Hermitian positive definiteness.

Applied Hermitian tensors to study holomorphic sectional curvature.

Abstract

We introduce $\hat{H}$-eigenvalue for $2m$-th order $n$-dimensional complex tensors. Then we determine several checkable inclusion sets for $\hat{H}$-eigenvalues and derive some criterions for the Hermitian positive definiteness (semi-definiteness) of Hermitian and CPS tensors. We also apply the Hermitian tensors to study holomorphic sectional curvature in complex differential geometry and reprove the algebraic part of recent results by Alvarez-Heier-Zheng and Chaturvedi-Heier.