The Density Theorem for Operator-Valued Frames via Square-Integrable Representations of Locally Compact Groups

Abstract

In this paper, we first prove a density theorem for operator-valued frames via square-integrable representations restricted to closed subgroups of locally compact groups, which is a natural extension of the density theorem in classical Gabor analysis. More precisely, it is proved that for such an operator-valued frame, the index subgroup is co-compact if and only if the generator is a Hilbert–Schmidt operator. Then we present some applications of this density theorem, and in particular establish necessary and sufficient conditions for the existence of such operator-valued frames with Hilbert–Schmidt generators. We also introduce the concept of wavelet transform for Hilbert–Schmidt operators, and use it to prove that if the representation space is infinite-dimensional, then the system indexed by the entire group is Bessel system but not a frame for the space of all Hilbert–Schmidt operators on the representation space.