Schur's Lemma for Coupled Reducibility and Coupled Normality

Let $\mathcal{A}=\{A_{ij}{\}}_{i,j\in \mathcal{I}}$, where $\mathcal{I}$ is an index set, be a doubly indexed family of matrices, where $A_{ij}$ is $n_i \times n_j$\mathrm{.}Foreach$i \in \mathcal I$,let$\mathcal V_i$bean$n_i$-dimensionalvectorspace\mathrm{.}Wesay$\mathcal A$ is reducible in the coupled sense if there exist subspaces, ${\mathcal{U}}_i\subseteq {\mathcal{V}}_i$, with ${\mathcal{U}}_i\mathrm{\ne }\{0\}$ for at least one $i\in \mathcal{I}$, and $\mathcal U_i \neq \mathcal V_i$foratleastone$i$,suchthat$A_{ij} (\mathcal U_j) \subseteq \mathcal U_i$forall$i, j$\mathrm{.}Let$\mathcal B = {B_{ij} }{i, j \in \mathcal I}$alsobeadoublyindexedfamilyofmatrices,where$B{ij}$is$m_i \times m_j$\mathrm{.}Foreach$i \in \mathcal I$,let$X_i$beamatrixofsize$n_i \times m_i$\mathrm{.}Suppose$A_{ij} X_j = X_i B_{ij}$forall$i, j$. We prove versions of Schur’s Lemma for $\mathcal{A},\mathcal{B}$ satisfying coupled irreducibility conditions. We also consider a refinement of Schur’s Lemma for sets of normal matrices and prove corresponding versions for $\mathcal A, \mathcal B$ satisfying coupled normality and coupled irreducibility conditions.