Upper bounds on the capacity of vector Gaussian channels affected by fading are derived under peak amplitude constraints at the input. The focus is on constraint regions that can be decomposed in a Cartesian product of sub-regions. This constraint models a transmitter configuration employing a number of power amplifiers less than or equal to the total number of transmitting antennas. In general, the power amplifiers feed distinct subsets of the transmitting antennas and partition the input in independent subspaces. Two upper bounds are derived: The first one is suitable for high signal-to-noise ratio (SNR) values and, as we prove, it is tight in this regime; The second upper bound is accurate at low SNR. Furthermore, the derived upper bounds are applied to the relevant case of amplitude constraints induced by employing a distinct power amplifier for each transmitting antenna.