Range spaces of co-analytic Toeplitz operators
William T. Ross,
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.
Toeplitz operator, Hardy space, range space, de Branges-Rovnyak space, boundary behavior, kernel function, non-extreme point, corona pair
30J05 - Inner functions
30H10 - Hardy spaces
46E22 - Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]