Composition operators within singly generated composition C*-algebras

Abstract

Abstract. Let ϕ be a linear-fractional self-map of the open unit disk D, not an automorphism, such that ϕ(ζ) = η for two distinct points ζ, η in the unit circle ∂D. We consider the question of which composition operators lie in C * (Cϕ, K), the unital C * -algebra generated by the composition operator Cϕ and the ideal K of compact operators, acting on the Hardy space H 2 . This necessitates a companion study of the unital C * -algebra generated by the composition operators induced by all parabolic non-automorphisms with common fixed point on the unit circle.

Compact Differences of Composition Operators in Several Variables

“…Specifically, if τ 1 is conjugate to translation by b, then τ 2 is conjugate to translation by c = b/|ψ (1)|; see Lemma 5 in [9]. Since b = 0, so also c = 0.…”

“…Many properties of composition operators have been studied over the past four decades; the monographs [4] and [14] give an overview of the work before the mid-1990s. Recently there has been considerable interest in studying algebras of composition operators, often modulo the ideal of compact operators (see, for example, [6][7][8][9][10]). In this direction, the question of when a difference C ϕ − C ψ is compact naturally arises.…”

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space H 2

“…In this paper, we use the results of T. L. Kriete and J. L. Moorhouse [11] and T. L. Kriete, B. D. MacCluer and J. L. Moorhouse [10] in order to investigate the essential normality problem for certain finite linear combinations of linear-fractional composition operators on H 2 .…”

“…Suppose that ϕ and w are as in Theorem 3.1 and F (ϕ) is as in Equation (10). In accordance with Equation (11), for each 0 i < n we assume that ϕ(ζ ri ) = ϕ(ζ ri+1 ) = .…”

Toeplitz-composition $C^{*}$-algebras for certain finite Blaschke products

“…Its quotient algebra by the ideal K of the compact operators is denoted by OC ϕ . Recently Kriete, MacCluer and Moorhouse [12,13] studied the Toeplitz-compostion C * -algebra T C ϕ for a certain linear fractional self-map ϕ. They describe the quotient C * -algebra OC ϕ concretely as a subalgebra of C(Λ) ⊗ M 2 (C) for a compact space Λ.…”