Spectra of linear fractional composition operators onH²(B_N)

Abstract

We characterize the spectra of composition operators on the Hardy space H 2 (B N ), when the symbols are elliptic or hyperbolic linear fractional self-maps of B N . Therefore, combining with the result obtained by Bayart [4], the spectra of all linear fractional composition operators on H 2 (B N ) are completely determined.

Commutators of automorphic composition operators with adjoints

“…Thus, the weighted Dirichlet space D s (B N ), in fact, is the weighted Hardy space H 2 (β, B N ) with the weight β(k) = (k + 1) (1−s)/2 . Using Proposition 2.4 of [13], we see that all linear fractional self-maps induce bounded composition operators on D s (B N ).…”

Essential normality of automorphic composition operators

“…For linear fractional self-maps of B N , the situation is more complicated. Especially, from the discussion of spectral structures of linear fractional composition operators on H 2 (B N ) (see [2], [3], [13]), we have found that linear fractional maps of B N , conjugated by automorphisms, must be classified into nine different cases. Until now, we only know a little about which linear fractional composition operators are essentially normal on the space H (see [14] and [17]).…”

“…At last, we give another class of linear fractional self-maps of B N whose corresponding composition operators are not essentially normal on H. The idea comes from the proofs of Proposition 2 in [17] and Theorem 2.3 in [13].…”

“…Let ϕ n denote the n-th iterate of ϕ. First, the proof of Theorem 2.3 in [13] gives ||ϕ n || ∞ < 1 for all n ≥ M , where M is a positive integer. Let m = min{n : ||ϕ n || ∞ < 1}.…”