Invariant subspaces of the harmonic Dirichlet space with large co-dimension

In this paper, we comment on the complexity of the invariant subspaces (under the bilateral Dirichlet shift f --> zeta f) of the harmonic Dirichlet space D. Using the sampling theory of Seip and some work on invariant subspaces of Bergman spaces, we will give examples of invariant subspaces F subset of D with dim(F/zeta F) = n, n is an element of N boolean OR{infinity}. We will also generalize this to the Dirichlet classes D-alpha, 0 < alpha < infinity, as well as the Besov classes B-p(alpha), 1 < p < infinity, 0 < alpha < 1.