Essential spectra of quasi-parabolic composition operators on Hardy spaces of analytic functions

Abstract

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as "quasi-parabolic." This is the class of composition operators on H 2 with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form ϕ(z) = z + ψ(z), where ψ ∈ H ∞ (H) and (ψ(z)) > > 0.We especially examine the case where ψ is discontinuous at infinity. A new method is devised to show that this type of composition operator fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.

Quasi-parabolic Composition Operators on Weighted Bergman Spaces

“…This paper is a continuation of our work [6] on quasi-parabolic composition operators on the Hardy space H 2 . In this work we investigate the same class of operators on weighted Bergman spaces A 2 α (D).…”

“…Proof. See [6] We also need the following lemma which characterizes certain integral operators as Fourier multipliers…”

“…Quasi-parabolic composition operators on H 2 (D) are composition operators induced by the symbols where 'a' is replaced by a bounded analytic function 'ψ' for which ℑ(ψ(z)) > δ > 0 ∀z ∈ D. We recall that the local essential range R ξ (ψ * ) of ψ ∈ H ∞ (D) at ξ ∈ T is defined to be the set of points ζ ∈ C for which the set {z ∈ T :| ψ * (z) − ζ |< ε} ∩ S ξ,r has positive Lebesgue measure ∀ε > 0 and ∀r > 0 where S ξ,r = {z ∈ T :| z − ξ |< r} and ψ * ∈ L ∞ (T) is the boundary value function of ψ. In [6] we showed that if ψ ∈ QC(T) ∩ H ∞ (D) then these composition operators are essentially normal and their essential spectra are given as…”

Essential Spectrum and Fredholm Index for Certain Composition Operators

“…The spectra and essential spectra of composition operators on spaces of analytic functions on the open unit disk have been studied by a number of authors (see [10], [9], [8], [3], [4]). Composition operators have also been studied in the context of uniform algebras (see [7], [5], [6]), where they arise as the unital homomorphisms of the algebras.…”

Essential spectra of quasi-parabolic composition operators on Hardy spaces of the Poly-disc

“…In this paper we study the essential spectra of a class of composition operators on the Hilbert-Hardy space of the bi-disc which is called "quasi-parabolic" and whose one variable analogue was studied in [2]. As in [2], quasi-parabolic composition operators on the Hilbert-Hardy space of the bi-disc are written as a linear combination of Toeplitz operators and Fourier multipliers.…”