Inner functions as multipliers and zero sets in weighted Dirichlet spaces

In this note, using some conditions on the weight function K, we investigate the inner functions as multipliers in weighted Dirichlet spaces, and we also discuss zero sets.

Integral operator acting on weighted Dirichlet spaces to Morrey type spaces

In this paper, we studied the boundedness and compactedness of integral operators from weighted Dirichlet spaces DK to Morrey type spaces H2K. Carleson measure and essential norm were also considered.

Range Spaces of Co-Analytic Toeplitz Operators

In this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.