Uniqueness of spectral flow

To each path of self-adjoint Fredholm operators acting on a real separable Hilbert space H with invertible ends, there is associated an integer called spectral flow. The purpose of this brief note is to show that spectral flow is uniquely characterized by four elementary properties: normalization, continuity, additivity over direct sums, and its value as the difference of the Morse indices of the ends when H is finite dimensional. The proof of uniqueness relies of the invariance of spectral flow of the path under cogredient transformations of the path.