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

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 {φ_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 {Jφ_s(. , .)}_{s\in Z}.