On infinite triangular matrices which are commutable with a banded matrix

Given a banded matrix Ar)with the diagonal elements all different and any r subdiagonals we are looking for all triangular matrices Xr)which commute with it. We show first that Xr)can be factorized in the form = Xr)Tr)-1diagμ)Tr)where the matrix Tr)and its inverse being given explicitely. Using this factorization we get an explicit expression for the elements of the matrix Xr)and also the set of eigenvalues and eigenspaces of these matrices. In particular we get an explicit expression for the elements of the inverse matrix of a normal matri Ar).