  We investigate pseudogap phenomena in a two-dimensional Fermi gas. Including
pairing fluctuations within a self-consistent $T$-matrix approximation, we
determine the pseudogap temperature $T^*$ below which a dip appears in the
density of states $\rho(\omega)$ around the Fermi level. Evaluating $T^*$, we
identify the pseudogap region in the phase diagram of this system. We find
that, while the observed BKT (Berezinskii-Kosterlitz-Thouless) transition
temperature $T^{\rm exp}_{\rm BKT}$ in a $^6$Li Fermi gas is in the pseudogap
regime, the detailed pseudogap structure in $\rho(\omega)$ at $T^{\rm exp}_{\rm
BKT}$ still differs from a fully-gapped one, indicating the importance of
amplitude fluctuations in the Cooper channel there. Since the observed $T^{\rm
exp}_{\rm BKT}$ in the weak-coupling regime cannot be explained by the recent
BKT theory which only includes phase fluctuations, our results may provide a
hint about how to improve this BKT theory. Although $\rho(\omega)$ has not been
measured in this system, we show that the assessment of our results is still
possible by using the observable Tan's contact.
