  We present the dynamical spin structure factor of the antiferromagnetic
spin-$\frac{1}{2}$ $J_1-J_2$ Heisenberg model on a triangular lattice obtained
from large-scale matrix-product state simulations. The high frustration due to
the combination of antiferromagnetic nearest and next-to-nearest neighbour
interactions yields a rich phase diagram. We resolve the low-energy excitations
both in the $120^{\circ}$-ordered phase and in the putative spin liquid phase
at $J_2/J_1 = 0.125$. In the ordered phase, we observe an avoided decay of the
lowest magnon-branch, demonstrating the robustness of this phenomenon in the
presence of gapless excitations. Our findings in the spin-liquid phase chime
with the field-theoretical predictions for a gapless Dirac spin liquid, in
particular the picture of low-lying monopole excitations at the corners of the
Brillouin zone. We comment on possible practical difficulties of distinguishing
proximate liquid and solid phases based on the dynamical structure factor.
