  Functions in backward shift invariant subspaces have nice analytic
continuation properties outside the spectrum of the inner function defining the
space. Inside the spectrum of the inner function, Ahern and Clark showed that
under some distribution condition on the zeros and the singular measure of the
inner function, it is possible to obtain non-tangential boundary values of
every function in the backward shift invariant subspace as well as for their
derivatives up to a certain order. Here we will investigate, at least when the
inner function is a Blaschke product, the non-tangential boundary values of the
functions of the backward shift invariant subspace after having applied a
co-analytic (truncated) Toeplitz operator. There appears to be a smoothing
effect.
