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

Abstract

Abstract. We compute the spectra and the essential spectra of bounded linear fractional composition operators acting on the Hardy and weighted Bergman spaces of the upper halfplane. We are also able to extend the results to weighted Dirichlet spaces of the upper halfplane. IntroductionThe Boundedness of a composition operator C τ on the Hardy or the weighted Bergman spaces of the half-plane has been proved to be equivalent with the angular derivative of the inducing map at infinity, denoted by τ ′ (∞), being finite and positive (see [14,6]). In addition, Elliott et al. [6,5] have shown that whenever C τ is bounded, the operator norm, the essential operator norm and the spectral radius are all equal and determined by the quantity τ ′ (∞). The above properties show that the spaces H 2 (Π + ) and A 2 α (Π + ) differ significantly from their unit disc analogues H 2 (D) and A 2 α (D) regarding the composition operators acting on them. We refer e.g. to [21,2] for expositions of the rich theory of composition operators on spaces defined on the unit disc.It is natural to ask what the spectral properties of composition operators acting on the halfplane are. 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. Indeed, C τ is bounded only when τ is a parabolic or a hyperbolic self-map of Π + fixing infinity. In this paper we compute the spectra and the essential spectra of these composition operators.In the parabolic case (see Theorem 3.1 in Section 3) we obtain the following result:Theorem A. Let τ be a parabolic self-map of Π + , that is, τ (w) = w + w 0 , where Im w 0 ≥ 0 and w 0 = 0. Then the spectrum of C τ acting on the Hardy or the weighted Bergman spaces of the upper half-plane equals2010 Mathematics Subject Classification. 47B33. Key words and phrases. Composition operator, spectrum, essential spectrum, Hardy space of the upper halfplane, weighted Bergman spaces of the upper half-plane.The author is supported by the Magnus Ehrnrooth Foundation in Finland.

Spectra of Linear Fractional Composition Operators on the Growth Space and Bloch Space of the Upper Half-Plane

“…We will answer this question affirmatively in the second section. Among composition operators, those induced by linear fractional transformations are well understood in several backgrounds, including the unit disk [3,5,6,[12][13][14]16] and the upper half-plane [11,20]. Recall that a linear fractional transformation (LFT) is a meromorphic bijection of the extended complex plane C ∪ {∞} onto itself, which can be expressed in the form…”

“…For details, see Sharpiro's book [21] or the article [12]. Recently, Schroderus [20] considered the spectrum problem of a linear fractional composition operator on H 2 (Π + ) and A 2 α (Π + ) and got a complete solution. Schroderus's result extended some earlier work of Gallardo-Gutiérrez and Montes-Rodríguez [11].…”

“…Taking the notation from [20] for the weighted Bergman space A 2 γ (Π + ), γ > −1, we associate A 2 −1 (Π + ) = H 2 (Π + ) and let K γ w be the reproducing kernel for…”

“…Obviously, ϕ is an automorphism of the upper half-plane when Im w 0 = 0. In this case, the author in [20] obtained the spectrum of C ϕ on the Hardy and weighted Bergman space of the upper half-plane by performing the Fourier transform to convert C ϕ into some multiplier on a certain L 2 space (Paley-Wiener theorem). The method cannot be applied to the growth space.…”