Variations on two Cabrelli's works

In this paper we present two different problems within the framework of shift-invariant theory. First, we develop a triangular form for shift-preserving operators acting on finitely generated shift-invariant spaces. In case of the normal operators, we recover a diagonal decomposition. The results show, in particular, that any finitely generated shift-invariant space can be decomposed into an orthogonal sum of principal shift-invariant spaces, with additional invariance properties under a shift-preserving operator. Second, we provide a new characterization of the multi-tiling sets $\Omega\subset\mathbb{R}^d$ of positive measure for which $L^2(\Omega)$ admits a structured Riesz basis of exponentials that is formulated in the ambient space $\mathbb{T}^{k\times k}$. In addition, we show a simpler sufficient condition which generalizes the admissibility property, that is also necessary for 2-tiling sets.