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 \(\dim \mathcal {N}/z\mathcal {N}=1\).