Pseudo-differential operators, Wigner transform and Weyl transform on the Similitude group, [formula omitted

Abstract

In this paper we study pseudo-differential operators with operator valued symbols on the Similitude group, SIM(2). We characterize the unique infinite-dimensional, unitary, irreducible representations of SIM(2) and using that the Fourier inversion formula is given. The operator valued Wigner transform and the Weyl transform are also studied here. Furthermore we also obtain a necessary and sufficient condition on the symbols for which the pseudo-differential operators SIM(2) are Hilbert-Schmidt and trace class operators. Finally, we also give the trace formula for these operators.