Composition operators on Hardy spaces of a half-plane

Abstract

We prove that a composition operator is bounded on the Hardy space H2 of the right half‐plane if and only if the inducing map fixes the point at infinity non‐tangentially, and has a finite angular derivative λ there. In this case the norm, essential norm and spectral radius of the operator are all equal to λ.

Similarity to an isometry of composition operators on the half-plane

“…That (1) ⇒ (2) is of course trivial, and we have already seen as a special case of Proposition 3.2 that (2) implies r is inner. By Corollary 3.5 in [3], we see that the second condition in (3) is equivalent to the statement that C r has norm 1. Since C n r = C r n for composition operators on the half plane, again by Corollary 3.5 in [3], if C r = 1, we would have a sequence f n ∈ H p (C + ) with each f n having norm 1 such that C n r f n → ∞ or C n r f n → 0 as n → ∞, which would contradict the Szőkefalvi-Nagy condition (Theorem 3.1).…”

Norm Estimates for Weighted Composition Operators on Spaces of Holomorphic Functions

“…In [7] it is shown that a composition operator C ϕ on H 2 (C + ) is bounded if and only if n.t.lim z→∞ ϕ(z) = ∞ and λ := n.t.lim z→∞ z/ϕ(z) exists (the angular derivative) with 0 < λ < ∞. Here n.t.lim denotes the nontangential limit, taken with z restricted to the sector …”

Invertible and normal composition operators on the Hilbert Hardy space of a half–plane

“…While boundedness of composition operators on H 2 .… C / was characterized in terms of Carleson measures [9], that criterion is considered hard to use in practice and so, a practical boundedness criterion, the norm, essential norm, and spectral radius formulas can be found in [5] (see also [12]): …”