Generalized Tensor Function via the Tensor Singular Value Decomposition based on the T-Product

In this paper, we present the definition of generalized tensor function according to the tensor singular value decomposition (T-SVD) based on the tensor T-product. Also, we introduce the compact singular value decomposition (T-CSVD) of tensors, from which the projection operators and Moore-Penrose inverse of tensors are obtained. We establish the Cauchy integral formula for tensors by using the partial isometry tensors and apply it into the solution of tensor equations. Then we establish the generalized tensor power and the Taylor expansion of tensors. Explicit generalized tensor functions are listed. We define the tensor bilinear and sesquilinear forms and propose theorems on structures preserved by generalized tensor functions. For complex tensors, we established an isomorphism between complex tensors and real tensors. In the last part of our paper, we find that the block circulant operator establishes an isomorphism between tensors and matrices. This isomorphism is used to prove the F-stochastic structure is invariant under generalized tensor functions.