In the present paper, we introduce new tensor krylov subspace methods for solving large Sylvester tensor equations. The proposed method uses the well-known T-product for tensors and tensor subspaces. We introduce some new tensor products and the related algebraic properties. These new products will enable us to develop third-order the tensor FOM (tFOM), GMRES (tGMRES), tubal Block Arnoldi and the tensor tubal Block Arnoldi method to solve large Sylvester tensor equation. We give some properties related to these method and present some numerical experiments.

翻译：本文提出了一种新的张量Krylov子空间方法，用于求解大规模Sylvester张量方程。该方法利用了著名的张量T-积及张量子空间理论。我们引入了一些新的张量乘积及其相关代数性质，这些新乘积使我们能够发展三阶张量FOM（tFOM）、张量GMRES（tGMRES）、管状块Arnoldi方法及张量管状块Arnoldi方法，以求解大规模Sylvester张量方程。我们给出了这些方法的相关性质，并呈现了数值实验结果。