How good is an isotropic gaussian input on a MIMO Ricean channel?

For a MIMO Ricean fading channel with perfect side information at the receiver we derive an analytic upper bound on the difference between capacity and the mutual information that is induced by an isotropic Gaussian input. We show that if the number of receiver antennas is at least equal to the number of transmitter antennas, then, as the signal-to-noise ratio tends to infinity, such an input is asymptotically optimal. But otherwise such an isotropic input might be suboptimal. We also propose an iterative algorithm to calculate the optimal power allocation.