On the $\eta$-function for bisingular pseudodifferential operators

TL;DR

This paper investigates the $\eta$-invariant for bisingular pseudodifferential operators, establishing trace properties, residue relations, and regularity at zero, advancing understanding of their spectral and algebraic structures.

Contribution

It introduces the trace property for the Wodzicki residue of bisingular operators, relates residues of the $\eta$-function to Wodzicki traces, and computes the $K$-theory of their algebra.

Findings

Proved the trace property for the Wodzicki residue of bisingular operators.

Expressed residues of the $\eta$-function in terms of Wodzicki trace of projections.

Established regularity of the $\eta$-function at zero for certain bisingular operators.

Abstract

In this work we consider the $\eta$-invariant for pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators. We study complex powers of classical bisingular operators. We prove the trace property for the Wodzicki residue of bisingular operators and show how the residues of the $\eta$-function can be expressed in terms of the Wodzicki trace of a projection operator. Then we calculate the $K$-theory of the algebra of $0$-order (global) bisingular operators. With these preparations we establish the regularity properties of the $\eta$-function at the origin for global bisingular operators which are self-adjoint, elliptic and of positive orders.