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

In the first part of the paper, we prove that a twisted shift-invariant subspace of $L^2({ℝ}^{2n})$ can be decomposed 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. Later, we introduce twisted shift preserving operators and the corresponding range operators. 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. Finally, we prove that the frame operator and its inverse associated with a system of twisted translates of ${\{{\varphi }_s\}}_{s\in \mathbb{Z}}$ are shift preserving and investigate their corresponding range operators.