In this paper Gabor system of certain type based on the unitary dual of the Heisenberg group $\mathbb{ H}^n$ is introduced and a sufficient condition is obtained for the Gabor system to be a Bessel sequence for $L^2(\mathbb{ R}^*,\mathcal{ B}_2;d\kappa)$ using the $Schr\"{ o}dinger$ representation of $\mathbb{ H}^n$, where $\mathcal{ B}_2$ denotes the class of Hilbert-Schmidt operators on $L^2(\mathbb{ R}^n)$ and $d\kappa$ denotes the Haar measure on $\mathbb{ R}^*$. Further a necessary and sufficient condition is provided for the Gabor system to be an orthonormal system, a Parseval frame sequence, a frame sequence and a Riesz sequence.
Categories: math.FA