Matrix-valued Gabor frames over LCA groups for operators

avruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely T-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operatorT. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L-2(G, C-nxn), where a bounded linear operator Theta on L-2(G, C-nxn) controls not only lower but also the upper frame condition. We term such frames matrix-valued (Theta, Theta*)-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 (Theta, Theta*)- Gabor frames in terms of hyponormal operators. It is shown that if Theta is adjointable hyponormal operator, then L2(G, Cnxn) admits a lambda-tight ( Theta, Theta*)-Gabor frame for every positive real number lambda. A characterization of matrix-valued (Theta, Theta*)-Gabor frames is given. Finally, we show that matrix-valued (Theta, Theta*)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.