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A generalized winding number formula for the Witten index of a Toeplitz operator

Masaki Izumi

TL;DR

This work generalizes the classical winding-number formula for the Fredholm index of Toeplitz operators to the Witten index ind_W T_f by establishing a principal-value unit-circle integral representation, ind_W T_f = -(1/2π i) p.v. ∮ f'(t)/f(t) dt, under mild regularity assumptions and finite zeros of f. The authors derive explicit trace formulae for Tr(φ(T_f^*T_f) − φ(T_fT_f^*)) using the Helton-Howe framework, including both boundary-circle and domain-integral forms, and connect these results to spectral-shift theory. They further extend these trace identities to operator monotone φ via Peller’s Besov-space criteria, obtaining compactness and trace-class results for a wide class of functions, with concrete expressions for Tr(|T_f|^p − |T_f^*|^p) and their boundary representations. Examples demonstrate the richness of the Witten-index values (including any real value) and illustrate the applicability to analytic and Besov-regular symbols. Overall, the paper strengthens the link between Toeplitz operator indices, spectral-shift concepts, and operator-monotone functional calculus in non-smooth/analytic settings.

Abstract

We generalize the winding number formula for the Fredholm index of a Toeplitz operator to the Witten index. We also show trace formulae involving Toeplitz operators and operator monotone functions.

A generalized winding number formula for the Witten index of a Toeplitz operator

TL;DR

This work generalizes the classical winding-number formula for the Fredholm index of Toeplitz operators to the Witten index ind_W T_f by establishing a principal-value unit-circle integral representation, ind_W T_f = -(1/2π i) p.v. ∮ f'(t)/f(t) dt, under mild regularity assumptions and finite zeros of f. The authors derive explicit trace formulae for Tr(φ(T_f^*T_f) − φ(T_fT_f^*)) using the Helton-Howe framework, including both boundary-circle and domain-integral forms, and connect these results to spectral-shift theory. They further extend these trace identities to operator monotone φ via Peller’s Besov-space criteria, obtaining compactness and trace-class results for a wide class of functions, with concrete expressions for Tr(|T_f|^p − |T_f^*|^p) and their boundary representations. Examples demonstrate the richness of the Witten-index values (including any real value) and illustrate the applicability to analytic and Besov-regular symbols. Overall, the paper strengthens the link between Toeplitz operator indices, spectral-shift concepts, and operator-monotone functional calculus in non-smooth/analytic settings.

Abstract

We generalize the winding number formula for the Fredholm index of a Toeplitz operator to the Witten index. We also show trace formulae involving Toeplitz operators and operator monotone functions.
Paper Structure (7 sections, 19 theorems, 133 equations)

This paper contains 7 sections, 19 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The Helton-Howe formula for Toeplitz operators
  5. Trace formulae for $\operatorname{Tr}(\varphi(T_f^*T_f)-\varphi(T_fT_f^*))$
  6. The Witten index formula
  7. Extension of the trace formulae to operator monotone functions

Key Result

Lemma 2.1

For $f,g\in W_2^{1/2}(\mathbb{T})\cap W_1^1(\mathbb{T})$, we have

Theorems & Definitions (45)

  • Lemma 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 35 more