The expressions of the generalized inverses of the block tensor via the C-Product

In this paper, we present the expressions of the generalized inverses of the third-order 2 x 2 block tensor under the C-Product. Firstly, we give the necessary and sufficient conditions to present some generalized inverses and the Moore-Penrose inverse of the block tensor in Banachiewicz-Schur forms. Next, some results are generalized to the group inverse and the Drazin inverse. Moreover, equivalent conditions for the existence as well as the expressions for the core inverse of the block tensor are obtained. Finally, the results are applied to express the quotient property and the first Sylsvester identity of tensors.