On the Index of Invariant Subspaces in the Space of Weak Products of Dirichlet Functions

Let denote the classical Dirichlet space of analytic functions in the open unit disc with finite Dirichlet integral, . Furthermore, let be the space of weak products of functions in , i.e. all functions that can be written as for some with . The dual of has been characterized in 2010 by Arcozzi, Rochberg, Sawyer, and Wick as the space of analytic functions on such that is a Carleson measure for the Dirichlet space. In this paper we show that for functions in proper weak*-closed -invariant subspaces of , the functions are in the Nevanlinna class of and have meromorphic pseudocontinuations in the Nevanlinna class of the exterior disc. We then use this result to show that every nonzero -invariant subspace of has index 1, i.e. satisfies dim N/z N = 1.