In this paper, we comment on the complexity of the invariant subspaces (under the bilateral Dirichlet shift f → ζf ) 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 ⊂ D with dim(F /ζF ) = n, n ∈ N ∪ {∞}. We will also generalize this to the Dirichlet classes Dα, 0 < α < ∞, ...

The main purpose of this article is to give an approach to the recent invariant subspace theorem of Brown, Chevreau and Pearcy: Every contraction on a Hubert space, whose spectrum contains the unit circle has nontrivial invariant subspaces. Our proof incorporates several of the recent ideas tying together function theory and operator theory. 1. I N T R O D U C T I O N The Jordan structure theor...

abstract The notion of invariant subspaces is useful in a number of theoretical and practical applications. In this paper we give an elementary treatment of invariant subspaces that stresses their connection with simple eigenvalues and their eigenvectors.