This paper is a well-written, well-organized and well-illustrated. It makes a valuable contribution in the study of the index on families of self-adjoint Fredholm operators. The author extends the notion of the index on Fredholm operators to the situation of families in which the parameter space is a finite disjoint union of compact connected spaces. This falls under the study of the (Atiyah-Patodi-Singer) spectral flow. Reviewer's remark: In my opinion, it would be interesting to investigate the case of families of (Spin) Dirac operators and to provide a topological realization for the analytical index given by the author, analogous to the well-known index theorem for families due to Atiyah-Singer; this may done through proving Riemann-Roch properties.