  Motivated by recent experimental measurements, we study the quasiparticle
spectra and the collective excitations in doped Sr$_2$IrO$_4$, in which the
interesting interplay between the electronic correlations and strong
spin-orbital coupling (SOC) exists. To include the SOC, we use the Hugenholtz
diagrams to extend the fluctuation exchange (FLEX) approach to the case where
the SU(2) symmetry can be broken. By using this generalized FLEX method, we
find a weak pseudogap behavior near $(\pi,0)$ in the slightly electron-doped
system, with the corresponding Fermi arc formed by the partial destruction of
Fermi surface. Similar features also appear in the hole-doped system, however,
the position of the Fermi arc is rotated $45^\circ$ with respect to the former.
These results are consistent with the recent angle-resolved photoemission
spectra in Sr$_2$IrO$_4$. We demonstrate that these anomalous features are
mainly caused by the isospin fluctuations derived from the effective
$J_{\text{eff}}=1/2$ doublet.
