  In this paper, the preconditioned TBiCOR and TCORS methods are presented for
solving the Sylvester tensor equation. A tensor Lanczos
$\mathcal{L}$-Biorthogonalization algorithm (TLB) is derived for solving the
Sylvester tensor equation. Two improved TLB methods are presented. One is the
biconjugate $\mathcal{L}$-orthogonal residual algorithm in tensor form
(TBiCOR), which implements the $LU$ decomposition for the triangular
coefficient matrix derived by the TLB method. The other is the conjugate
$\mathcal{L}$-orthogonal residual squared algorithm in tensor form (TCORS),
which introduces a square operator to the residual of the TBiCOR algorithm. A
preconditioner based on the nearest Kronecker product is used to accelerate the
TBiCOR and TCORS algorithms, and we obtain the preconditioned TBiCOR algorithm
(PTBiCOR) and preconditioned TCORS algorithm (PTCORS). The proposed algorithms
are proved to be convergent within finite steps of iteration without roundoff
errors. Several examples illustrate that the preconditioned TBiCOR and TCORS
algorithms present excellent convergence.
