Groups of unitary composition operators on Hardy-Smirnov spaces

Abstract

Let Ω be an open simply connected proper subset of the complex plane. We identify, up to isomorphism, which groups are possible for the group of unitary composition operators of a Hardy-Smirnov space defined on Ω. We also study the relationship between the geometry of Ω and the corresponding group.

Composition operators on Hardy-Smirnov spaces

“…∈ Ω to determine which symbols Φ and Ψ induce Hermitian and unitary composition operators, respectively (see, for instance, [1,5,17,18]). Here we shall introduce a different approach.…”

“…For the case of normal operators (see [25,Theorem 5.1.15]), composition operators on H 2 (U) are induced by Φ(z) = λz with |λ| ≤ 1. The characterization of unitary composition operators on Hardy-Smirnov spaces was given by Gunatillake, Jovovic and Smith in [18,Theorem 3.2]: C Φ is a unitary bounded composition operator on H 2 (Ω) if, and only if, there are complex numbers λ and r with |λ| = 1 such that Φ(z) = λz + r is an automorphism of Ω. The case of Hermitian composition operators on H 2 (Ω) was studied by Gunatillake in [17,Theorem 3.3], where it was shown that if C Φ is a Hermitian bounded composition operator, then Φ(z) = λz + r for some real number λ.…”

“…In contrast with complex symmetric composition operators on H 2 (U), Hai and Severiano ([20,Theorem 8.8]) showed that if Φ is an analytic self-map of C + with a fixed point in C + then C Φ is never complex symmetric on H 2 (C + ). Since unitary and Hermitian bounded operators are complex symmetric, [17,Theorem 3.3] and [18,Theorem 3.2] provide natural ways to obtain examples of complex symmetric composition operators on H 2 (Ω). However, it is not immediate with respect to which conjugations these operators are complex symmetric.…”

“…Using these expressions, in Section 4, we describe all composition operators whose adjoints are also composition operators (Theorem 4.2) and we characterize the Hermitian and unitary composition operators on H 2 (Ω) (Theorems 4.7 and 4.9). In particular, we show how the numbers λ and r given in [17,Theorem 3.3] and [18,Theorem 3.2] depend on the coefficients of τ. We also provide concrete examples of normal composition operators on H 2 (Ω).…”