Weighted numerical range and weighted numerical radius for even-order tensor via Einstein product

The main aim of this article is to introduce the weighted numerical range and the weighted numerical radius for an even-order square tensor via the Einstein product and establish their various properties. Also, the proof of convexity of the numerical range of a tensor is revisited. The notions of weighted unitary tensor, weighted positive definite tensor, and weighted positive semi-definite tensor are then discussed. The spectral decomposition for normal tensors is also provided. This is then used to present the equality between the weighted numerical radius and the spectral radius of a weighted normal tensor. As applications of the above fact, a few equalities of weighted numerical radius and weighted tensor norm are obtained.