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

Abstract

Let $D$ denote the classical Dirichlet space of analytic functions $f$ in the open unit disc $\mathbb {D}$ with finite Dirichlet integral, $\int _\mathbb {D}|f'|^2 dA < \infty$. Furthermore, let $D \odot D$ be the space of weak products of functions in $D$, i.e. all functions $h$ that can be written as $h = \sum _{i=1}^\infty f_i g_i$ for some $f_i, g_i \in D$ with $\sum _{i=1}^\infty \Vert f_i\Vert \Vert g_i\Vert < \infty$. The dual of $D \odot D$ has been characterized in 2010 by Arcozzi, Rochberg, Sawyer, and Wick as the space $\mathcal {X}(D)$ of analytic functions $b$ on $\mathbb {D}$ such that $|b'|^2 dA$ is a Carleson measure for the Dirichlet space. In this paper we show that for functions $f$ in proper weak*-closed $M_z^*$-invariant subspaces of $\mathcal {X}(D)$, the functions $(zf)'$ are in the Nevanlinna class of $\mathbb {D}$ and have meromorphic pseudocontinuations in the Nevanlinna class of the exterior disc. We then use this result to show that every nonzero $M_z$-invariant subspace $\mathcal {N}$ of $D \odot D$ has index 1, i.e. satisfies $\dim \mathcal {N}/z\mathcal {N}=1$.