  It is shown that an elliptic scattering operator $A$ on a compact manifold
with boundary with coefficients in the bounded operators of a bundle of Banach
spaces of class (HT) and Pisier's property $(\alpha)$ has maximal regularity
(up to a spectral shift), provided that the spectrum of the principal symbol of
$A$ on the scattering cotangent bundle of the manifold avoids the right
half-plane.
  This is deduced directly from a Seeley theorem, i.e. the resolvent is
represented in terms of pseudodifferential operators with R-bounded symbols,
thus showing by an iteration argument the R-boundedness of
$\lambda(A-\lambda)^{-1}$ for $\Re(\lambda) \geq 0$.
  To this end, elements of a symbolic and operator calculus of
pseudodifferential operators with R-bounded symbols are introduced. The
significance of this method for proving maximal regularity results for partial
differential operators is underscored by considering also a more elementary
situation of anisotropic elliptic operators on $R^d$ with operator valued
coefficients.
