The regularity of the $\eta$ function for the Shubin calculus

Abstract

We prove the regularity of the $\eta$ function for classical pseudodifferential operators with Shubin symbols. We recall the construction of complex powers and of the Wodzicki and Kontsevich-Vishik functionals for classical symbols on ${\mathbb{R}}^n$ with these symbols. We then define the $\zeta$ and $\eta$ functions associated to suitable elliptic operators. We compute the $K_0$ group of the algebra of zero-order operators and use this knowledge to show that the Wodzicki trace of the idempotents in the algebra vanishes. From this, it follows that the $\eta$ function is regular at 0 for any self-adjoint elliptic operator of positive order.