In this paper we study the transport operator given by the sum of a free-streaming term and of an absorption term. The velocity coefficient appearing in the definition of the operator is unbounded ($v\in {\RR}$) while the position coordinate belongs to the one-dimensional bounded set $[-b, +b]$. The boundary condition associated to the transport operator are given by means of a bounded and positive operator $\Lambda$. We prove that an evolution problem associated to the above transport operator has a unique positive solution whose explicit form is given by means of a $C_0$-semigroup in the case $\Vert \Lambda \Vert \leq 1$ and by means of an integrated semigroup or a $C$-semigroup in the case $\Vert \Lambda \Vert >1$.