tensor product $(\mathbb{K}^d)^{\otimes q}$.

For two $\mathbb{K}$-valued tensors $S,T\in\mathcal{T}^q_d(\mathbb{K})$, we define their Hadamard-type product (i.e. entry-wise product) by $$S\circ T:=(S_{i_1,\dots,i_q}T_{i_1,\dots,i_q})_{1\leq i_1,\dots,i_q\leq d}\in\mathcal{T}_d^q(\mathbb{K}).$$ Also, we set $$\|T\|_{\ell_1}:=\sum_{1\leq i_1,\dots,i_q\leq d}|T_{i_1,\dots,i_q}|,\qquad
\|T\|_{\ell_\infty}:=\max_{1\leq i_1,\dots,i_q\leq d}|T_{i_1,\dots,i_q}|.$$

For $T\in \mathcal{T}^q_d(V)$ and $v\in V$, we define $$\langle T,v\rangle_V:=(\langle T_{i_1,\dots,i_q},v\rangle_V)_{1\leq i_1,\dots,i_q\leq d}\in\mathcal{T}^q_d(\mathbb{K}).$$

Let $H_1,H_2$ be two real separable Hilbert space. For $S\in \mathcal{T}^{p}_d(H_1)$ and $T\in \mathcal{T}^{q}_d(H_2)$, we define $$S\otimes T=(S_{i_1,\dots,i_p}\otimes T_{j_1,\dots,j_q})_{1\leq i_1,\dots,i_p,j_1,\dots,j_q\leq d}\in\mathcal{T}^{p+q}_d(H_1\otimes H_2).$$

Multi-way arrays {#sec:array}
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In this subsection we introduce some notation related to multi-way arrays (or tensors) which are necessary to state our main results.

Given a positive integer $N$, we set $[N]:=\{1,\dots,N\}$ for short. We denote by $\mathbb{K}$ the real field $\mathbb{R}$ or the complex field $\mathbb{C}$ and consider a vector space $V$ over $\mathbb{K}$. Given $q$ positive integers $N_1,\dots,N_q$, we denote by $V^{N_1\times\cdots\times N_q}$ the set of all $V$-valued $N_1\times\cdots\times N_q$ arrays, i.e. $V$-valued functions on $[N_1]\times\cdots\times[N_q]$. Note that $V^{N_1\times N_2}$ corresponds to the set of all $V$-valued $N_1\times N_2$ matrices. When $N_1=\cdots=N_q=N$, we call an element of $V^{N_1\times\cdots\times N_q}$ a $V$-valued $N$-dimensional $q$-way array. For an array $T\in V^{N_1\times\cdots\times N_q}$ and indices $i_k\in [N_k]$ ($k=1,\dots,q$), we write $T(i_1,\dots,i_q)$ as $T^{i_1,\dots,i_q}$ and $T$ itself as $T=(T^{i_1,\dots,i_q})_{(i_1,\dots,i_q)\in\prod_{k=1}^q[N_k]}$. When $V=\mathbb{K}$, $V^{N_1\times\cdots\times N_q}$ is naturally 