Matrix-Valued Gabor Frames over LCA Groups for Operators

Abstract

G\v avruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely $K$-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator $K$. For a locally compact abelian group G and a positive integer $n$, we study frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space $L^2(G, \mathbb{C}^{n\times n})$ , where a bounded linear operator $Θ$ on $L^2(G, \mathbb{C}^{n\times n})$ controls not only lower but also the upper frame condition. We term such frames matrix-valued $(Θ, Θ^*)$-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued $(Θ, Θ^*)$- Gabor frames in terms of hyponormal operators. It is shown that if $Θ$ is adjointable hyponormal operator, then $L^2(G, \mathbb{C}^{n\times n})$ admits a $λ$-tight $(Θ, Θ^*)$-Gabor frame for every positive real number $λ$. A characterization of matrix-valued $(Θ, Θ^*)$-Gabor frames is given. Finally, we show that matrix-valued $(Θ, Θ^*)$-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.