  We consider the behavior of an ultracold Fermi gas across a narrow Feshbach
resonance, where the occupation of the closed channel may not be negligible.
While the corrections to the single-channel formulae associated with the
nonzero chemical potential and with particle conservation have been considered
in the existing literature, there is a further effect, namely the
"inter-channel Pauli exclusion principle" associated with the fact that a
single hyperfine species may be common to the two channels. We focus on this
effect and show that, as intuitively expected, the resulting corrections are of
order $E_F/\eta$, where $E_F$ is the Fermi energy of the gas in the absence of
interactions and $\eta$ is the Zeeman energy difference between the two
channels. We also consider the related corrections to the fermionic excitation
spectrum, and briefly discuss the collective modes of the system.
