Localised frames for tensor product spaces
Dimitri Bytchenkoff, Michael Speckbacher, Peter Balazs
TL;DR
The paper addresses whether tensoring two self-localised frames preserves localisation. It develops an inverse-closed rank-four tensor Banach algebra $\mathcal{A}$ by intersecting two solid spectral matrix algebras and equipping it with a doubly contracted product; the key requirement is that the operator-valued versions $\widetilde{\mathcal{A}}_1, \widetilde{\mathcal{A}}_2$ are inverse-closed. Under this framework, the Gram tensor of the tensor product frame $G_{\Psi_1 \otimes \Psi_2}$ lies in $\mathcal{A}$, implying the tensor product frame remains self-localised with respect to the constructed tensor algebra. The results extend to frames of Hilbert–Schmidt operators and offer guidance on handling anisotropic vs isotropic tensorizations, with solidity not being essential for the core co-orbit conclusions. Overall, the work provides a practical pathway to apply co-orbit theory to tensor product frames and broadens the class of inverse-closed algebras applicable to localisation analysis.
Abstract
In this paper, we investigate whether the tensor product of two frames, each individually localised with respect to a spectral matrix algebra, is also localised with respect to a suitably chosen tensor product algebra. We provide a partial answer by constructing an involutive Banach algebra of rank-four tensors that is built from two solid spectral matrix algebras. We show that this algebra is inverse-closed, given that the original algebras satisfy a specific property related to operator-valued versions of these algebras. This condition is satisfied by all commonly used solid spectral matrix algebras. We then prove that the tensor product of two self-localised frames remains self-localised with respect to our newly constructed tensor algebra. Additionally, we discuss generalisations to localised frames of Hilbert-Schmidt operators, which may not necessarily consist of rank-one operators.