Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 029, 21 pages      arXiv:2508.01061      https://doi.org/10.3842/SIGMA.2026.029
Contribution to the Special Issue on Asymptotics, Randomness and Noncommutativity

On the Uniqueness of the $G$-Equivariant Spectral Flow

Marek Izydorek a, Joanna Janczewska a, Maciej Starostka b and Nils Waterstraat b
a) Institute of Applied Mathematics, Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland
b) Martin-Luther-Universität Halle-Wittenberg, Naturwissenschaftliche Fakultät II, Institut für Mathematik, 06099 Halle (Saale), Germany

Received August 06, 2025, in final form March 02, 2026; Published online March 25, 2026

Abstract
The spectral flow is an integer-valued homotopy invariant for paths of selfadjoint Fredholm operators. Lesch as well as Pejsachowicz, Fitzpatrick and Ciriza independently showed that it is uniquely characterised by its elementary properties. The authors recently introduced a $G$-equivariant spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the action of a compact Lie group $G$. The purpose of this paper is to show that the $G$-equivariant spectral flow is uniquely characterised by the same elementary properties when appropriately restated. As an application, we introduce an alternative definition of the $G$-equivariant spectral flow via a $G$-equivariant Maslov index.

Key words: spectral flow; equivariant linear operators; symplectic Hilbert spaces; Maslov index.

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