Some Properties of Range Operators on LCA Groups

Ruchika Verma ; Kumari Teena

Kragujevac Journal of Mathematics, Tome 47 (2023) no. 7, p. 995 .

Résumé

In this paper, we study the structure of shift preserving operators acting on shift-invariant spaces in $L^{2}(G)$, where $G$ is a locally compact Abelian group. We generalize some results related to shift-preserving operator and its associated range operator from $L^{2}(\mathbb{R}^{d})$ to $L^{2}(G)$. We investigate the matrix structure of range operator $R(\xi) $ 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 $L^{2}(G)$. 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 $R^{-1}(\xi)$ of range operator on $L^{2}(G)$.

Détail

DOI : 10.46793/KgJMat2307.0995V

Classification : 47A15, 22B99, 42C15, 43A25
Keywords: Shift-invariant space, range function, range operator, locally compact Abelian group, shift preserving operator, frame, Parseval frame, normal operator, unitary operator