  We theoretically investigate excitation properties in the pseudogap regime of
a trapped Fermi gas. Using a combined $T$-matrix theory with the local density
approximation, we calculate strong-coupling corrections to single-particle
local density of states (LDOS), as well as the single-particle local spectral
weight (LSW). Starting from the superfluid phase transition temperature $T_{\rm
c}$, we clarify how the pseudogap structures in these quantities disappear with
increasing the temperature. As in the case of a uniform Fermi gas, LDOS and LSW
give different pseudogap temperatures $T^*$ and $T^{**}$ at which the pseudogap
structures in these quantities completely disappear. Determining $T^*$ and
$T^{**}$ over the entire BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein
condensate) crossover region, we identify the pseudogap regime in the phase
diagram with respect to the temperature and the interaction strength. We also
show that the so-called back-bending peak recently observed in the
photoemission spectra by JILA group may be explained as an effect of pseudogap
phenomenon in the trap center. Since strong pairing fluctuations, spatial
inhomogeneity, and finite temperatures, are important keys in considering real
cold Fermi gases, our results would be useful for clarifying normal state
properties of this strongly interacting Fermi system.
