Solution Analysis of Tensor Equation $\mathcal{A} \ltimes \mathcal{X} \ltimes \mathcal{B}= \mathcal{C}$ via Semi Tensor Product with t-product

This paper focuses on the analysis of the tensor equation $\mathcal{A\ltimes X\ltimes B=C}$, formulated via the semi tensor product with t-product.

For the unknown vector $\mathcal{X}$, we establish a necessary and sufficient condition that provides an equivalence criterion for the existence of solutions.

For matrix valued and higher-order tensor valued unknown $\mathcal{X}$, solvability is determined by corresponding compatibility requirements.

Moreover, the explicit structure (Toeplitz and Circulant) of $\mathcal{C}$ is characterized.

The derived results are supported by several illustrative examples.