Wodzicki residue for operators on manifolds with cylindrical ends

We define the Wodzicki Residue TR(A) for A belonging to a space of operators with double order, denoted \({L^{m_1,m_2}_{\rm cl}}\). Such operators are globally defined initially on \({\mathbb{R}^n}\) and then, more generally, on a class of non-compact manifolds, namely, the manifolds with cylindrical ends. The definition is based on the analysis of the associate zeta function ζ(A, z). Using this approach, under suitable ellipticity assumptions, we also compute a two terms leading part of the Weyl formula for a positive selfadjoint operator \({A\in L^{m_1,m_2}_{\rm cl}}\) in the case m ₁ = m ₂.