Schur's Lemma for Coupled Reducibility and Coupled Normality

Abstract

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 $n_i\times n_j$. For each $i\in \mathcal{I}$, let ${\mathcal{V}}_i$ be an $n_i$-dimensional vector space. We say $\mathcal{A}$ is reducible in the coupled sense if there exist subspaces, ${\mathcal{U}}_i\subseteq {\mathcal{V}}_i$, with ${\mathcal{U}}_i\ne \{0\}$ for at least one $i\in \mathcal{I}$, and ${\mathcal{U}}_i\ne {\mathcal{V}}_i$ for at least one $i$, such that $A_{ij}({\mathcal{U}}_j)\subseteq {\mathcal{U}}_i$ for all $i,j$. Let $\mathcal{B}=\{B_{ij}{\}}_{i,j\in \mathcal{I}}$ also be a doubly indexed family of matrices, where $B_{ij}$ is $m_i\times m_j$. For each $i\in \mathcal{I}$, let $X_i$ be a matrix of size $n_i\times m_i$. Suppose $A_{ij}{X}_j=X_i{B}_{ij}$ for all~$i,j$. 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.

35 pages. Second version corrects some typos in the original submission and makes some changes in MSC classification numbers