  We investigate strong-coupling properties of a trapped two-dimensional normal
Fermi gas. Within the framework of a combined $T$-matrix theory with the local
density approximation, we calculate the local density of states, as well as the
photoemission spectrum, to see how two-dimensional pairing fluctuations affect
these single-particle quantities. In the BCS (Bardeen-Cooper-Schrieffer)-BEC
(Bose-Einstein condensation) crossover region, we show that the local density
of states exhibits a dip structure in the trap center, which is more remarkable
than the three-dimensional case. This pseudogap phenomenon is found to
naturally lead to a double peak structure in the photoemission spectrum. The
peak-to-peak energy of the spectrum at p=0 agrees well with the recent
experiment on a two-dimensional 40K Fermi gas [M. Feld, et al., Nature 480, 75
(2011)]. Since pairing fluctuations are sensitive to the dimensionality of a
system, our results would be useful for the study of many-body physics in the
BCS-BEC crossover regime of a two-dimensional Fermi gas.
