  Utilizing the unbiased time-dependent density-matrix renormalization group
technique, we examine the photoemission spectra in the extended Falicov-Kimball
model at zero and finite temperatures, particularly with regard to the
excitonic insulator state most likely observed in the quasi-one-dimensional
material Ta$_2$NiSe$_5$. Working with infinite boundary conditions, we are able
to simulate all dynamical correlation functions directly in the thermodynamic
limit. For model parameters best suited for Ta$_2$NiSe$_5$ the photoemission
spectra show a weak but clearly visible two-peak structure, around the Fermi
momenta $k\simeq\pm k_{\rm F}$, which suggests that Ta$_2$NiSe$_5$ develops an
excitonic insulator of BCS-like type. At higher temperatures, the leakage of
the conduction-electron band beyond the Fermi energy becomes distinct, which
provides a possible explanation for the bare non-interacting band structure
seen in time- and angle-resolved photoemission spectroscopy experiments.
