Linear-Fractional Composition Operators in Several Variables

Abstract

We investigate properties of linear-fractional composition operators Cϕ on Hardy and Bergman spaces of the ball in C N that are motivated by a formula for the self-commutator [C * ϕ , Cϕ]. In particular, we characterize when certain commutators [Cϕ, Cσ] are compact, and give conditions under which [T * z β , Cϕ] is compact, where T z β is multiplication by the monomial z β . Our results allow us to determine when Cϕ is essentially normal, for ϕ belonging to a large class of linear-fractional symbols.

Compact Differences of Composition Operators in Several Variables

“…A special case of Theorem 4 appears in [11], and the work in [11] has been extended in [5]. Our techiques here are different, and rely on an extension to several variables of the key geometric arguments of the last section, which we turn to next.…”

“…Our starting point, in either the disk or the ball, will be the following necessary condition for compactness of C ϕ − C ψ . Theorem 1 [11,12]. Suppose ϕ, ψ are holomorphic self-maps of D (respectively, B N ) and suppose that there exists a sequence of points z n tending to the boundary of D (B N ) along which ρ(ϕ(z n ), ψ(z n )) 1 − |z n | 2 1 − |ϕ(z n )| 2 + 1 − |z n | 2 1 − |ψ(z n )| 2 (1) does not converge to zero, where ρ(ϕ(z n ), ψ(z n )) is defined by…”

On a Class of Composition Operators on Bergman Space

“…The problem is subtle and no general answer is known. In this work, we have shown that the Berezin symbol of a bounded linear operator S from the Bergman space into itself satisfies certain averaging condition if and only if the operator S satisfy the intertwining relation C a SC a = S for all a ∈ D. Recently, the spectra of composition operators have attracted much attention (see [5][6][7]) from operator theorists. To this purpose, it is important to know what are the essential commutants of the invertible operators C a ,a ∈ D, and to characterize those…”

“…In this work, we present a necessary and sufficient condition for C a SC a − S = 0 to happen for all a ∈ D in terms of the Berezin symbol of S. Related work in this area can be found in [5][6][7][8].…”

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

“…P. S. Bourdon, D. Levi, S. K. Narayan, and J. H. Shapiro in [3] have shown that a composition operator induced on H 2 by a linear-fractional self-map of the unit disk is nontrivially essentially normal if and only if it is induced by a parabolic non-automorphism. The essentially normal composition operators on other spaces have been investigated by some authors (see, e.g., [4], [12], and [13]). If ϕ and ψ are linear-fractional self-maps of D or B N , then C ϕ − C ψ cannot be non-trivially compact; i.e., if the difference is compact, either C ϕ and C ψ are individually compact or ϕ = ψ.…”