We investigate the structure of invariant subspaces of backward shift operator Lf = (f − f(0))/ζ on a large class of abstract Hilbert spaces of analytic functions on the unit disc where the forward shift operator Mζf = ζf acts as a contraction. Our main results show that under certain regularity conditions on the norm of such a space, the functions in a nontrivial invariant subspace of L have m...

1. Introduction. The research reported on in the present note was motivated by the following Proposition (F), due to Ernest Fischer ([5], see also [4] for an earlier version; actually Fischer proved a more general result, but the special case suffices as a point of departure for our discussion) : (F) Let P denote a homogeneous polynomial in si, • • • , z k with complex coefficients. Then every ...