  In this note we study the structure of shift-preserving operators acting on a
finitely generated shift-invariant space. We define a new notion of
diagonalization for these operators, which we call s-diagonalization. We give
necessary and sufficient conditions on a bounded shift-preserving operator in
order to be s-diagonalizable. These conditions are in terms of its range
operator. We also obtain a generalized Spectral Theorem for normal bounded
shift-preserving operators.
