Some Properties of Range Operators on LCA Groups

In this paper, we study the structure of shift preserving operators acting on shift-invariant spaces in L2(G), where G is a locally compact Abelian group. We generalize some results related to shift-preserving operator and its associated range operator from L2(ℝd) to L2(G). We investigate the matrix structure of range operator R(ξ) on range function J associated to shift-invariant space V , in the case of a locally compact Abelian group G. We also focus on some properties like as normal and unitary operator for range operator on L2(G). We show that shift preserving operator U is invertible if and only if fiber of corresponding range operator R is invertible and investigate the measurability of inverse R−1(ξ) of range operator on L2(G).