Localization Operator and Weyl Transform on Reduced Heisenberg Group with Multi-dimensional Center

Abstract

In this article, we study two different types of operators, the localization operator and Weyl transform, on the reduced Heisenberg group with multidimensional center $\mathcal{G}$. The group $\mathcal{G}$ is a quotient group of non-isotropic Heisenberg group with multidimensional center ${\mathcal{H}}^m$ by its center subgroup. Firstly, we define the localization operator using a wavelet transform on $\mathcal{G}$ and obtain the product formula for the localization operators. Next, we define the Weyl transform associated to the Wigner transform on $\mathcal{G}$ with the operator-valued symbol. Finally, we have shown that the Weyl transform is not only a bounded operator but also a compact operator when the operator-valued symbol is in $L^p,1\le p\le 2,$ and it is an unbounded operator when $p>2$.