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We consider a continuous path of bounded symmetric Fredholm bilinear formswith arbitrary endpoints on a real Hilbert space, and we prove a formula thatgives the spectral flow of the path in terms of the spectral flow of therestriction to a finite codimensional closed subspace. We also discuss the caseof restrictions to a continuous path of finite codimensional closed subspaces.As an application of the formula, we introduce the notion of spectral flow fora periodic semi-Riemannian geodesic, and we compute its value in terms of theMaslov index.

On a formula for the spectral flow and its applications