We define pseudo-differential operators on \((0,\infty )\) as pseudo-differential operators on the locally compact, Hausdorff and abelian group \({\mathbb {R}}^+\) , where \({\mathbb {R}}^+\) is the group with the underlying set \((0,\infty )\) and the binary operation given by the multiplication of real numbers. Realizing \({\mathbb {R}}\) as the dual group of \({\mathbb {R}}^+\) , we define pseudo-differential operators on \({\mathbb {R}}^+\) with symbol in \(L^2({\mathbb {R}}^+\times {\mathbb {R}})\) by means of the Mellin transform. We give them explicit formulas for the symbols of the products and the adjoints, chararacterize them as Hilbert–Schmidt operators on \(L^2({\mathbb {R}}^+,\mu )\) , where \(\mu \) is the left and right Haar measure on \({\mathbb {R}}^+\) . We also characterize the ideal of trace class pseudo-differential operators in it in terms of the symbols lying in a subspace W of \(L^2({\mathbb {R}}^+\times {\mathbb {R}})\) and give a trace formula for all these trace class operators. In particular, we show that \(L^2({\mathbb {R}}^+\times {\mathbb {R}})\) is a \(H^*\) -algebra of functions on the group \({\mathbb {R}}^+\times {\mathbb {R}}\) and W is an ideal in the function algebra \(L^2({\mathbb {R}}^+\times {\mathbb {R}})\) .
