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.

Restriction of Toeplitz Operators on Their Reducing Subspaces

We study the restrictions of analytic Toeplitz operator on its minimal reducing subspaces for the unit disc and construct their models on slit domains. Furthermore, it is shown that Tzn is similar to the sum of n copies of the Bergman shift.

On Quasisimilarity for Analytic Toeplitz Operators

Let f be a function in H∞. We show that if f is inner or if the commutant of the analytic Toeplitz operator Tf is equal to that of Tb for some finite Blaschke product b, then any analytic Toeplitz operator quasisimilar to Tf is unitarily equivalent to Tf.

Analytic Toeplitz and Composition Operators

This paper is a continuation of [1] where we began the study of intertwining analytic Toeplitz operators. Recall that X intertwines two operators A and B if XA = BX. Let H2 be the Hilbert space of analytic functions in the open unit disk D for which the functions fr(θ) = f(reiθ) are bounded in the L2 norm, and H∞ be the set of bounded functions in H2. For φ ∊ Hφ, Tφ (or Tφ(z)) is the analytic Toeplitz operator defined on H2 by the relation (Tφf)(z) = φ(z)f(z). For φ ∊ H∞, we shall denote {φ(z): |z| < 1} by Range (φ) or φ(D). Then where and σ(Tφ) = Closure(φ(D)) [1]. If φ ∊ H∞ maps D into D, then we define the composition operator Cφ on H2 by the relation (Cφf) (z) = f(φ(z)). J. Ryff has shown [11, Theorem 1] that Cφ, is a bounded linear operator on H2.