  Considered are operators that leave the set of non-invertible (in the sense
of Ehrenpreis) distributions stable. They simultaneously generalise the
operation of convolution by a distribution with compact support and the
operation of multiplication by a real analytic function; they are here called
pseudo-convolutions since they also generalise pseudo-differential operators.
(It is shown that the elliptic real analytic pseudo-differential operators
leave both the non-invertible and the invertible distributions invariant.) But
when the condition of real-analyticity is relaxed, such operators may map a
non-invertible distribution to one invertible -- given that the invertibility
in both cases concerns the same function space. By varying the space, however,
one can measure the 'loss of non-invertibily' that a non-analytic perturbation
may introduce. This phenomenon is here studied using the Beurling classes of
functions and measuring the regularity of operator symbols in the
Denjoy-Carleman sense; the Gevrey case turns out particularly simple.
