The regularity of the $$\eta $$ η 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 all self-adjoint elliptic operator of positive order.