An order N tensor $\mathcal{A}={\left(a_{i_1\dots i_N}\right)}_{1⩽i_j⩽I_j}(j=1,\dots ,N)$ over a field $\mathbb{F}$ is a multidimensional array with $I_1{I}_2\dots I_N$ entries in $\mathbb{F}$, where N is a positive integer ([27,44]). The set of all such N tensors is denoted by ${\mathbb{F}}^{I_1\times \dots \times I_N}$. In the past decades, there have been active researches on tensors due to various applications in many areas such as physics, computer vision, data mining, etc. (see, e.g., [9,11,15,16,17,21,25,26,27,29,30,31,32,33,35,36,37,41,42,43,54,56]).

In this paper, we investigate the solvability of some tensor equations over quaternions. This is motivated by the recent researches on tensor equations as well as a long research history of solving matrix equations, which are briefly outlined as follows. It is well-known that the following Sylvester matrix equation and its generalized forms have been widely investigated and found numerous applications in many areas.

During the past decades, many methods have been developed for solving Sylvester-type matrix equation over the quaternion algebra. For example, Kyrchei [28] gave explicit determinantal representation formulas of solutions to the Equation (1). Heyouni et al. [20] presented the SGl-CMRH method and preconditioned framework of this method to solve matrix Equation (1) when $X=Y$. Zhang [53] investigated the general system of generalized Sylvester quaternion matrix equations. Ahmadi-Asl and Beik [1,2,7] presented effective iterative algorithms for solving different quaternion matrix equations. Song [47] investigated the general solution to a system of quaternion matrix equation by Cramer’s rule. Wang et al. [48] considered the solvability of a system of constrained two-sided coupled generalized Sylvester quaternion matrix equations. Zhang et al. [52] obtained some special least squares solutions of the quaternion matrix equation $AXB+CXD=E$. Huang et al. [22] considered the modified conjugate gradient method to solve the generalized coupled Sylvester conjugate matrix equations.

Quaternions offer greater versatility and flexibility compared to real and complex numbers, particularly in addressing multidimensional problems. This unique property has attracted growing interest among scholars, leading to numerous valuable achievements in quaternion-related research (see, e.g., [13,19,23,34,38,50,51]). The tensor equation is a natural extension of the matrix equation.

In this paper, we consider the following generalized Sylvester tensor equation over $\mathbb{H}$: where the tensors $A^{\left(n\right)}$, $B^{\left(n\right)}\in {\mathbb{H}}^{I_n\times I_n}(n=1,2,\dots ,N)$, $\mathcal{C}\in {\mathbb{H}}^{I_1\times \dots \times I_N}$ are given, and the tensors $\mathcal{X}$, $\mathcal{Y}\in {\mathbb{H}}^{I_1\times \dots \times I_N}$ are unknown. The n-mode product of a tensor $\mathcal{X}\in {\mathbb{H}}^{I_1\times \dots \times I_N}$ with a matrix $A\in {\mathbb{H}}^{I_n\times I_n}$ is defined as Note that the n-mode product can also be expressed in terms of unfolded quaternion tensors: where ${\mathcal{X}}_{\left[n\right]}$ is the mode-n unfolding of $\mathcal{X}$ ([27]). We consider the following problems corresponding to (2):

Problem 1.1: For given the tensor $\mathcal{C}\in {\mathbb{H}}^{I_1\times I_2\times \dots \times I_N}$, the matrices $A^{\left(n\right)},B^{\left(n\right)}\in {\mathbb{H}}^{I_n\times I_n}(n=$$1,2,\dots ,N)$, find the tensors $\tilde{\mathcal{X}},\tilde{\mathcal{Y}}\in {\mathbb{H}}^{I_1\times I_2\times \dots \times I_N}$ such that

Problem 1.2: Let the solution set of Problem 1.1 be denoted by $S_{XY}$. For a given tensor ${\mathcal{X}}_0,{\mathcal{Y}}_0\in {\mathbb{H}}^{I_1\times I_2\times \dots \times I_N}$, find the tensor $\check{\mathcal{X}},\check{\mathcal{Y}}\in {\mathbb{H}}^{I_1\times I_2\times \dots \times I_N}$ such that

It is worth emphasizing that the tensor equation (2) includes several well-studied matrix/tensor equations as special cases. For example, if $\mathcal{X}$ and $\mathcal{Y}$ in (2) are order 2 tensors, i.e., matrices, then Equation (2) can be reduced to the following extended Sylvester matrix equation In the case of $B^{\left(n\right)}=0(n=1,2\dots ,N)$, Equation (2) becomes the following equation which has been studied extensively in recent years. For instance, Saberi et al. [45,46] investigated the SGMRES-BTF method and SGCRO-BTF method to solve Equation (3) over $\mathbb{R}$. Wang et al. [49] gave the conjugate gradient least squares method to solve Equation (3) over $\mathbb{H}$. Zhang and Wang [55] presented the tensor forms of the bi-conjugate gradient (BiCG-BTF) and bi-conjugate residual (BiCR-BTF) methods for solving tensor Equation (3) over real number field $\mathbb{R}$. Chen and Lu [10] considered a projection method with a Kronecker product preconditioner to solve Equation (3) over $\mathbb{R}$. Karimi and Dehghan [24] presented the tensor form of global least squares method for finding an approximate solution of (3). Furthermore, Najafi-Kalyani et al. [39] derived some iterative algorithms on global Hessenberg process in their tensor forms to solve Equation (3). Considering Equation (3) over $\mathbb{R}$ and $N=3$, that is, It has been shown that Equation (4) plays an important role in finite difference [3], thermal radiation [29], information retrieval [33], finite elements [14], and microscopic heat transport problem [37]. Therefore, our study on Equation (2) will provide a unified treatment for these matrix/tensor equations.

The rest of this paper is organized as follows. In Section 2, we recall some definitions and notations, and prove several lemmas for changing Equation (2). In Section 3, we develop the BiCG iterative algorithm for solving the quaternion tensor Equation (2), and prove that our algorithm is correct. We also show that the minimal Frobenious norm solution can be obtained by choosing special kinds of initial tensors. Some numerical examples are presented in Section 4 for illustrating the efficiency and applications of the proposed algorithm. Finally, we summarize our contributions in Section 5.

We first recall some notations and definitions. For two complex matrices $U=\left(u_{ij}\right)\in {\mathbb{C}}^{m\times n},V=\left(v_{ij}\right)\in {\mathbb{C}}^{p\times q}$, the symbol $U\otimes V=\left(u_{ij}V\right)\in {\mathbb{C}}^{mp\times nq}$ denotes the Kronecker product of U and V.

The operator $vec(·)$ is defined as: for a matrix A and a tensor $\mathcal{X}$, respectively, where $a_k$ is the kth column of A and ${\mathcal{X}}_{\left[1\right]}$ is the mode-1 unfolding of the tensor $\mathcal{X}$. The inner product of two tensors $\mathcal{X},\mathcal{Y}\in {\mathbb{H}}^{I_1\times \dots \times I_N}$ is defined by where ${\overline{y}}_{i_1{i}_2\dots i_N}$ represents the quaternion conjugate of $y_{i_1{i}_2\dots i_N}$. If $\langle \mathcal{X},\mathcal{Y}\rangle =0$, then we say that tensors $\mathcal{X}$ and $\mathcal{Y}$ are orthogonal. The Frobenius norm of tensor $\mathcal{X}$ is defined as $\parallel \mathcal{X}\parallel =\sqrt{\langle \mathcal{X},\mathcal{X}\rangle }.$

For any $\mathcal{X}\in {\mathbb{H}}^{I_1\times \dots \times I_N}$, it is well-known that $\mathcal{X}$ can be uniquely expressed as $\mathcal{X}={\mathcal{X}}_1+{\mathcal{X}}_2\mathbf{i}+{\mathcal{X}}_3\mathbf{j}+{\mathcal{X}}_4\mathbf{k}$, where ${\mathcal{X}}_i\in {\mathbb{R}}^{I_1\times \dots \times I_N},i=1,2,3,4$. Next we define n-mode operators for ${\mathcal{X}}_i$.

Let $A^{\left(n\right)}=A_1^{\left(n\right)}+A_2^{\left(n\right)}\mathbf{i}+A_3^{\left(n\right)}\mathbf{j}+A_4^{\left(n\right)}\mathbf{k},B^{\left(n\right)}=B_1^{\left(n\right)}+B_2^{\left(n\right)}\mathbf{i}+B_3^{\left(n\right)}\mathbf{j}+B_4^{\left(n\right)}\mathbf{k}\in {\mathbb{H}}^{I_n\times I_n}$, where $A_i^{\left(n\right)},B_i^{\left(n\right)}\in {\mathbb{R}}^{I_n\times I_n},i=1,2,3,4$. For $\mathcal{W}\in {\mathbb{R}}^{I_1\times \dots \times I_N}$, we define

Next, replacing $\mathcal{W}$ in above equations by ${\mathcal{X}}_i$’s, we define the following notations:

The following lemma shows that the quaternion tensor Equation (2) is equivalent to a system of four real tensor equations.

From Lemma 1 and 2, it is easy to see that the uniqueness of the solution of equation (2) can be described as follows:

Given fixed matrices $A^{\left(n\right)}\in {\mathbb{R}}^{I_n\times I_n},n=1,2,\dots ,N$, we define the following linear operators

Using the property $〈\mathcal{X},\mathcal{Y}{\times }_n{A}^{\left(n\right)}〉=〈\mathcal{X}{\times }_n{\left(A^{\left(n\right)}\right)}^{\mathrm{T}},\mathcal{Y}〉$ in [26], it is easy to prove the following lemma.

Clearly, ${\mathcal{L}}_{A^{\left(n\right)}}$ defined above is a linear mapping. The following lemma provides the uniqueness of the dual mapping for these kinds of linear mappings.

Finally, we use linear operators $\mathcal{L}$ and ${\mathcal{L}}^{*}$ to describe the inner products involving ${\Gamma }_i$ and ${\Phi }_i$, which we will use in next sections.