  Parameter-elliptic boundary-value problems are investigated on the extended
Sobolev scale. This scale consists of all Hilbert spaces that are interpolation
spaces with respect to the Hilbert Sobolev scale. The latter are the
H\"ormander spaces $B_{2,k}$ for which the smoothness index $k$ is an arbitrary
radial function RO-varying at infinity. We prove that the operator
corresponding to this problem sets isomorphisms between appropriate H\"ormander
spaces provided that the absolute value of the parameter is large enough. For
solutions to the problem, we establish two-sided estimates, in which the
constants are independent of the parameter.
