Decomposition of spectral flow and Bott-type iteration formula (Q2202472)

The paper is well written and of a certain interest in index theory for self-adjoint Fredholm operators. The main result of the paper is Theorem 1.3, which deals with a decomposition formula for the spectral flow of \textit{M. F. Atiyah} et al. [Math. Proc. Camb. Philos. Soc. 79, 71--99 (1976; Zbl 0325.58015)] associated to continuous paths of self-adjoint Fredholm operators which are invariant under cogredient-like matrices; this is a lift of the decomposition formula of \textit{X. Hu} and \textit{S. Sun} [Commun. Math. Phys. 290, No. 2, 737--777 (2009; Zbl 1231.37031)] to a more general case. A major ingredient in the proof of Theorem 1.3, in Section 3, is the direct sum property of spectral flow Lemma 2.2: it is used at level of the cogredient invariance property of spectral flow Lemma 1.1 (notice that Lemma 1.1 contents a generalization of a result of \textit{P. Fitzpatrick} et al. [``Spectral flow for paths of unbounded operators and bifurcation of critical points'', work in progress]). Using Theorem 1.3, the authors propose a Bott-type iteration formula extending a lot of well-known cases and moreover giving some new generalizations, see Sections 4.5. In particular, Bott-type iteration formulas of Maslov-index are given by exploiting the well-known identity between Maslov index and spectral flow.