  We investigate the large time behavior of solutions to the spatially
homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis
is performed in some (weighted) $L^{1}$-spaces. We deal with both the cases of
hard and soft potentials (with angular cut-off). For hard potentials, we
provide a new proof of the fact that, in weighted $L^{1}$-spaces with
exponential or algebraic weights, the solutions converge exponentially fast
towards equilibrium. Our approach uses weak-compactness arguments combined with
recent results of the second author on positive semigroups in $L^{1}$-spaces.
For soft potentials, in $L^{1}$-spaces, we exploits the convergence to ergodic
projection for perturbed substochastic semigroup to show that, for very general
initial datum, solutions to the linear Boltzmann equation converges to
equilibrium in large time. Moreover, for a large class of initial data, we also
prove that the convergence rate is at least algebraic. Notice that, for soft
potentials, no exponential rate of convergence is expected because of the
absence of spectral gap.
