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

TL;DR

This paper establishes the regularity of the $\eta$ function at zero for self-adjoint elliptic operators within the Shubin calculus, extending the understanding of spectral invariants for pseudodifferential operators.

Contribution

It proves the regularity of the $\eta$ function for classical pseudodifferential operators with Shubin symbols, including the construction of complex powers and trace functionals.

Findings

The $\eta$ function is regular at zero for self-adjoint elliptic operators.

The $K_0$ group of the algebra of zero-order operators is computed.

The Wodzicki trace of idempotents in the algebra vanishes.

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.