ABSTRACT. 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 [25] 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. We investigate in the present work the large time behavior of solutions to the linear Boltzmann equation for both hard and soft potentials, under some cut-off assumption. We consider solutions in some weighted L 1 -spaces and our approach is based on functional analytic results regarding positive semigroups in such spaces. To keep the presentation and results simple, we investigate only here the spatially homogeneous Boltzmann equation which exhibits already a quite rich behavior. Extension to spatially inhomogeneous problems is planned for future investigations.1.1. The kinetic model. Before entering the details of our results, let us formulate the problem we aim to address here: we shall consider the spatially homogeneous linear Boltzmann equation (BE in the sequel) ∂ t f (t, v) = Lf (t, v), f (0, v) = f 0 (v) 0 (1.1) in which L is the linear Boltzmann operator given bywhere Q(f, g) denotes the bilinear Boltzmann operatorwhere v ′ and v ′ * are the pre-collisional velocities which result, respectively, in v and v * after elastic collision(1.4)Here f and g are nonnegative functions of the velocity variable v ∈ R d and B(q, n) is a nonnegative function . We will assume throughout this paper that the distribution function M appearing in (1.2) is a given Maxwellian function: 5) where u ∈ R d is the given bulk velocity and Θ > 0 is the given effective temperature of the host fluid. Notice that, by Galilean invariance, there is no loss of generality in assumingWe shall investigate in this paper several collision operators L = L B corresponding to various interactions collision kernels B = B(v − v * , σ). Typically, we shall consider the casewhere b : [−1, 1] → R + and Φ : R + → R + are measurable. We shall consider models with angular cut-off, i.e. we assumewhich, in turns, readsNotice that, without loss of generality, we can assume that ℓ b = 1. Typically, we shall consider the case of power-like potentials Φ(·):where we distinguish between tw...