  We study the boundary-dissipated transverse field Ising model described by a
Lindblad Master equation and exactly solve its Liouvillian spectrum in the
whole parameter space. By mapping the Liouvillian into a Su-Schrieffer-Heeger
model with imaginary boundary potentials under a parity constraint, we solve
the rapidity spectrum analytically and thus construct the Liouvillian spectrum
strictly with a parity constraint condition. Our results demonstrate that the
Liouvillian spectrum displays four different structures, which are
characterized by different numbers of segments. By analyzing the properties of
rapidity spectrum, we can determine the phase boundaries between different
spectrum structures analytically and prove the Liouvillian gap fulfilling a
duality relation in the weak and strong dissipation region. Furthermore, we
unveil the existence of a dynamical duality, i.e., the long-time relaxation
dynamics exhibits almost the same dynamical behavior in the weak and strong
dissipation region as long as the duality relation holds true.
