  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 $\Theta$ 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 $(\Theta, \Theta^*)$-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 $L^2(G, \mathbb{C}^{n\times n})$ 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.
