  We propose a new viewpoint on Hilbert scales extending them by means of all
Hilbert spaces that are interpolation ones between spaces on the scale. We
prove that this extension admits an explicit description with the help of
$\mathrm{OR}$-varying functions of the operator generating the scale. We also
show that this extended Hilbert scale is obtained by the quadratic
interpolation (with function parameter) between the above spaces and is closed
with respect to the quadratic interpolation between Hilbert spaces. We give
applications of the extended Hilbert scale to interpolational inequalities,
generalized Sobolev spaces, and spectral expansions induced by abstract and
elliptic operators.
