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 flo w is uniquely characterized by four elementary properties: normalization, c ontinuity, 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 un iqueness relies of the invariance of spectral flow of the path under cogred ient transformations of the path. (C) 2000 Elsevier Science Ltd. All rights reserved.