  The spectral function of one hole in different magnetic states of the
one-dimensional t-J model including three-site term and frustration
$J^{\prime}$ is studied. In the strong coupling limit $J \to 0$ (corresponding
to $U \to \infty$ of the Hubbard-model) a set of eigenoperators of the
Liouvillian is found which allows to derive an exact expression for the
one-particle Green's function that is also applicable at finite temperature and
in an arbitrary magnetic state. The spinon dispersion of the pure t-J model
with the ground-state of the Heisenberg model can be obtained by treating the
corrections due to a small exchange term by means of the projection method. The
spectral function for the special frustration $J^{\prime}=J/2$ with the
Majumdar-Ghosh wave function is discussed in detail. Besides the projection
method, a variational ansatz with the set of eigenoperators of the $t$-term is
used. We find a symmetric spinon dispersion around the momentum $k=\pi/(2a)$
and a strong damping of the holon branch. Below the continuum a bound state is
obtained with finite spectral weight and a very small separation from the
continuum. Furthermore, the spectral function of the ideal paramagnetic case at
a temperature $k_B T \gg J$ is discussed.
