  We prove reducibility of a class of quasi-periodically forced linear
equations of the form \[ \partial_tu-\partial_x\circ (1+a(\omega t,
x))u+\mathcal{Q}(\omega t)u=0,\quad x\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z},
\] where $u=u(t,x)$, $a$ is a small, $C^{\infty}$ function, $\mathcal{Q}$ is a
pseudo differential operator of order $-1$, provided that
$\omega\in\mathbb{R}^{\nu}$ satisfies appropriate non-resonance conditions.
Such PDEs arise by linearizing the Degasperis-Procesi (DP) equation at a small
amplitude quasi-periodic function. Our work provides a first fundamental step
in developing a KAM theory for perturbations of the DP equation on the circle.
Following \cite{Airy}, our approach is based on two main points: first a
reduction in orders based on an Egorov type theorem then a KAM diagonalization
scheme. In both steps the key difficulites arise from the asymptotically linear
dispersion law. In view of the application to the nonlinear context we prove
sharp \emph{tame} bounds on the diagonalizing change of variables.
  We remark that the strategy and the techniques proposed are applicable for
proving reducibility of more general classes of linear pseudo differential
first order operators.
