Invertible and normal composition operators on the Hilbert Hardy space of a half–plane

Abstract

Operators on function spaces of form C f D f ı , where is a fixed map are called composition operators with symbol . We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

Hermitian composition operators on Hardy-Smirnov spaces

“…Proof of (3); since C is bounded and Hermitian ' is continuous on the closed unit disc (see Theorem B). Since 0 exist at p we let z D p in (8). Then r D ' 0 .p/: From part (6) of Lemma 4.1 it follows that ' 0 .p/ D 1:…”

“…To prove (2) let z D p in (8). Proof of (3); since C is bounded and Hermitian ' is continuous on the closed unit disc (see Theorem B).…”

“…/ are studied in [10]. Also, composition operators on a Hardy space of a half-plane are studied in [7], [8], and [9]. In this paper we study bounded Hermitian composition operators on H 2 .…”

Spectra of linear fractional composition operators on the Hardy and weighted Bergman spaces of the half-plane

“…Composition operators C τ : f → f • τ , where τ : Π + −→ Π + is analytic, acting on these spaces are, however, much less studied when compared to their counterparts in the unit disc setting. Matache [12] found a condition for boundedness of C τ on H 2 (Π + ) in terms of Carleson measures and showed later in [13] that there are no compact composition operators on H 2 (Π + ). In [22] Shapiro and Smith extended the non-compactness result to A 2 α (Π + ).…”

“…In fact, in the unit disc setting the spectral picture of composition operators has been completely determined when the inducing maps are linear fractional transformations and the operators act on weighted Dirichlet spaces, including H 2 (D) and A 2 α (D) for all α > −1 (see, for instance, [17,2,10,8,19,7]). The spectra in the corresponding setting on the half-plane are largely unknown; in the (unweighted) Dirichlet space of Π + the spectra are known but besides that only the spectra (and the essential spectra) for invertible or self-adjoint parabolic and invertible hyperbolic composition operators acting on the Hardy space H 2 (Π + ) have been computed (see [15]). In contrast to the unit disc case, not all linear fractional transformations τ induce bounded composition operators on H 2 (Π + ) or A 2 α (Π + ).…”