Twisted shift preserving operators on $L^{2}(\mathbb{R}^{2n})$

TL;DR

This paper introduces a new framework using the Zak transform to analyze twisted shift-invariant spaces on $L^{2}(\mathbb{R}^{2n})$, revealing their structure and properties of associated operators.

Contribution

It develops a decomposition of twisted shift-invariant subspaces into orthogonal principal spaces and studies properties of twisted shift preserving operators and their range operators.

Findings

Decomposition of twisted shift-invariant subspaces into orthogonal components.

Twisted shift preserving operators share properties like self-adjointness and unitarity.

Frame operators for twisted translates are shift preserving and linked to dual Gramian.

Abstract

We introduce the J map using the Zak transform associated with the Weyl transform on $L^{2}(\mathbb{R}^{2n})$. We obtain a decomposition for a twisted shift-invariant subspace of $L^{2}(\mathbb{R}^{2n})$ as a direct sum of mutually orthogonal principal twisted shift-invariant spaces such that the respective system of twisted translates forms a Parseval frame sequence. We establish that the twisted shift preserving operators and the corresponding range operators simultaneously share some properties in common, namely, self-adjoint, unitary, range of the spectrum and bounded below properties. We prove that the frame operator and its inverse associated with a system of twisted translates of {\varphi_s}_{s\in Z} are shift preserving. We also show that the corresponding range operators turn out to be the dual Gramian and its inverse associated with the collection {J\varphi_s(. , .)}_{s\in Z}.