Research Projects

This research proposal is centered around the study of Dirichlet-type spaces, both in one and several complex variables, with particular attention to the study of operator tuples that naturally act on these spaces. The project is organized into three primary components, each focusing on a different geometric setting: the unit disc, the bidisc, and the unit ball. Additionally, we consider possible generalizations to broader classes of domains where a suitable notion of Dirichlet-type space can be defined. In the single-variable case, the analysis is largely directed toward the wandering subspace problem, a well-known and unresolved issue in operator theory. A key open question in this context is whether a norm-increasing analytic 3-isometry possesses the wandering subspace property. This problem is of considerable interest, as its resolution could lead to significant advancements in the structure theory of operators on Dirichlet-type spaces. For the bidisc, the theory becomes richer and more intricate. There exist multiple, non-equivalent notions of Dirichlet-type spaces over the bidisc, each leading to distinct challenges in function theory and operator theory. Many foundational questions remain open. One of the central aims of this project is to address the Gleason problem for Dirichlet-type spaces over the full bidisc. A solution to this problem would have profound implications for the understanding of the structure of associated operator tuples. Additionally, we investigate the concept of left-inverse commutativity among operator tuples acting on these spaces. Understanding and classifying such tuples—particularly up to unitary equivalence—represents an independent and compelling direction of study. The investigation then moves to Dirichlet-type spaces on the unit ball in several complex variables. Compared to the disc and bidisc settings, these spaces are relatively less developed and understood. Foundational questions, such as whether the coordinate functions act as multipliers or whether polynomials are dense in these spaces, are still unresolved. Before we can meaningfully classify operator tuples those that are cyclic analytic joint 2-isometries, it is essential to determine which multiplication operator tuples qualify as joint 2-isometries—a question that lies at the intersection of function theory and multivariable operator theory. Finally, we aim to explore how these questions and techniques might extend to more general domains, provided an appropriate definition of Dirichlet-type spaces can be established. This includes domains such as the symmetrized bidisc and the Hartogs triangle, where Hardy spaces are well-studied, but corresponding Dirichlet-type spaces are not yet defined. Identifying or constructing appropriate analogs of Dirichlet-type spaces on such domains is not only desirable but necessary for the broader applicability of the theory. In summary, this proposal outlines a comprehensive study of Dirichlet-type spaces across multiple settings, aiming to tackle fundamental problems in function theory and multivariable operator theory, with potential applications to more abstract or geometrically complex domains.