On Almost-Type Special Structured Tensor Classes Associated with Semi-Positive Tensors

In this paper, we introduce almost (strictly) semi-positive tensors, which extend the concept of almost (strictly) semimonotone matrices. Furthermore, we provide insights into the characteristics of the entries within these almost (strictly) semi-positive tensors and establish a condition that is both necessary and sufficient for categorizing the underlying tensor as an almost semi-positive tensor. Drawing inspiration from H. V\"{a}liaho's work on copositivity, we present the concept of almost (strictly) copositive tensors, which extends the notion of almost (strictly) copositive matrices to tensors. It is shown that a real symmetric tensor is almost (strictly) semi-positive if and only if it is almost (strictly) copositive and a symmetric almost (strictly) semi-positive tensor has a (nonpositive) negative $H^{++}$-eigenvalue. We also establish a relationship between (strictly) diagonally dominant and (strong) $\mathcal{M}$-tensors with (strictly) semi-positive tensors.