Refined Geršhgorin disks for tensors and \(G{\mathcal {H}}\)-tensors

In this paper, exploiting the structure of tensors, the so called refined Geršhgorin disks for H-eigenvalues of tensors are introduced, which are always tighter than the classical Geršhgorin-type theorem for H-eigenvalues of tensors introduced by Qi (J Symb Comput 40:1302–1324, 2005). A sufficient condition for the positivity of even-order tensors is also given. We introduce the definition of \(G{\mathcal {H}}\) $G\mathcal{H}$ -tensors, which can be viewed as a generalization of \({\mathcal {H}}\) $\mathcal{H}$ -tensors. Moreover, an algorithm for identifying \(G{\mathcal {H}}\) $G\mathcal{H}$ -tensors is also obtained. We prove that the tensor equation \({\mathcal {A}}x^{m-1}={\textbf{b}}\) $\mathcal{A}{x}^{m-1}=\mathbf{b}$ has a unique positive solution if \({\mathcal {A}}\) $\mathcal{A}$ is a strong \(G{\mathcal {H}}^{+}\) $G{\mathcal{H}}^{+}$ -tensor and \({\textbf{b}}\in {\mathbb {R}}^n\) $\mathbf{b}\in {\mathbb{R}}^n$ is a positive vector.