Schur's Lemma for Coupled Reducibility and Coupled Normality

Let A = {A(ij)}(i,j is an element of I), where I is an index set, be a doubly indexed family of matrices, where A(ij) is n(i) x n(j). For each i is an element of I, let V-i be an n(i)-dimensional vector space. We say A is reducible in the coupled sense if there exist subspaces, U-i subset of V-i, with U-i not equal {0} for at least one i is an element of I, and u(i) not equal V-i for at least one i, such that A(ij)(U-j)( )subset of U-i for all i , j. Let B = {B-ij}(i,j is an element of I) also be a doubly indexed family of matrices, where B-ij is m(i) x m(j). For each i is an element of I, let X-i be a matrix of size n(i) x m(i). Suppose A(ij)X(j) = XiBij for all i, j. We prove versions of Schur's lemma for A, B satisfying coupled irreducibility conditions. We also consider a refinement of Schur's lemma for sets of normal matrices and prove corresponding versions for A, B satisfying coupled normality and coupled irreducibility conditions.