Several Trace Formulas for Auxiliary Operators on the Half Space

In this chapter, by using the Weyl–Hörmander calculus we prove two partial trace formulas on the half axis \({\mathbf {R}}_{+}\) (see Theorems 20.1 and 20.6). Table 20.1 gives a list of short time trace formulas for the three fundamental operators \(W_{0,0}(t)\) , \(W_{1,0}(t)\) and \(K_{1}(t)\) as \(t \downarrow 0\) .