A transformation rule for the index of commuting operators

TL;DR

This paper develops transformation rules for the index of commuting operators using Koszul complexes, providing global and local index theorems applicable to Toeplitz operators on various function spaces.

Contribution

It introduces new transformation rules for the index under holomorphic functional calculus, including global and local index theorems for commuting operators.

Findings

Global index expressed via degree and index functions

Local index computed near zeros of the symbol

Results applicable to Toeplitz operators on Bergman and Hardy spaces

Abstract

In the setting of several commuting operators on a Hilbert space one defines the notions of invertibility and Fredholmness in terms of the associated Koszul complex. The index problem then consists of computing the Euler characteristic of such a special type of Fredholm complex. In this paper we investigate transformation rules for the index under the holomorphic functional calculus. We distinguish between two different types of index results: 1) A global index theorem which expresses the index in terms of the degree function of the "symbol" and the locally constant index function of the "coordinates". 2) A local index theorem which computes the Euler characteristic of a localized Koszul complex near a common zero of the "symbol". Our results apply to the example of Toeplitz operators acting on both Bergman spaces over pseudoconvex domains and the Hardy space over the polydisc. The local index theorem is fundamental for future investigations of determinants and torsion of Koszul complexes.