On the boundedness of the support of optimal input measures for Rayleigh fading channels

We consider transmission over a wireless multiple antenna communication system operating in a Rayleigh flat fading environment with no channel state information at the receiver and the transmitter. We show that, subject to the average power constraint, the support of the capacity achieving input distribution is bounded. Moreover, we show by a simple example concerning the identity theorem (or uniqueness theorem) from the complex analysis in several variables that some of the existing results in the field are not rigorous.