Hankel Matrices Acting on the Dirichlet Space

Abstract

The study of the infinite Hankel matrix acting on analytic function spaces dates back to the influential work of Nehari and Widom on the Hardy space \(H^2\). Since then, it has been extensively generalized to other settings such as weighted Bergman spaces, Dirichlet type spaces, and Möbius invariant function spaces. Nevertheless, several fundamental operator-theoretic questions, including the boundedness and compactness, remain unresolved in the context of the Dirichlet space. Motivated by this, via Carleson measures, the Widom type condition, and the reproducing kernel thesis, we obtain:

1. (i)

   necessary and sufficient conditions for bounded and compact operators induced by Hankel matrices on the Dirichlet space, thereby answering a folk question in this field (Galanopoulos et al. in Result Math 78(3) Paper No. 106, 2023);

2. (ii)

   necessary and sufficient conditions for bounded and compact operators induced by Cesàro type matrices on the Dirichlet space.

As a beneficial product, we find an intrinsic function-theoretic characterization of functions with positive decreasing Taylor coefficients in the function space \({\mathcal {X}}\) throughly studied by Arcozzi et al. (Lond Math Soc II Ser 83(1):1–18, 2011). In addition, we also show that a random Dirichlet function almost surely induces a compact Hankel type operator on the Dirichlet space.