  The Bott index of two unitary operators on an infinite dimensional Hilbert
space is defined. homotopic invariance with respect to multiplicative unitary
perturbations of the type identity plus trace class and the "logarithmic" law
for the index are proven. The index and its properties are then extended to the
case of a pair of invertible operators. An application to the physics of two
dimensional quantum systems proves that the index is equal to the Chern number
therefore showing that the transverse Hall conductance is an integer.
