  By a famous result, functions in backward shift invariant subspaces in Hardy
spaces are characterized by the fact that they admit a pseudocontinuation a.e.
on $\T$. More can be said if the spectrum of the associated inner function has
holes on $\T$. Then the functions of the invariant subspaces even extend
analytically through these holes. We will discuss the situation in weighted
backward shift invariant subspaces. The results on analytic continuation will
be applied to consider some embeddings of weighted invariant subspaces into
their unweighted companions. Such weighted versions of invariant subspaces
appear naturally in the context of Toeplitz operators. A connection between the
spectrum of the inner function and the approximate point spectrum of the
backward shift in the weighted situation is established in the spirit of
results by Aleman, Richter and Ross.
