Complex symmetry and cyclicity of composition operators on $H^2(\mathbb {C}_+)$

Abstract

In this article, we completely characterize the complex symmetry, cyclicity and hypercyclicity of composition operators C φ f = f • φ induced by affine self-maps φ of the right half-plane C + on the Hardy-Hilbert space H 2 (C + ). The interplay between complex symmetry and cyclicity plays a key role in the analysis. We also provide new proofs for the normal, self-adjoint and unitary cases and for an adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodríguez.2010 Mathematics Subject Classification. Primary 47B33, 47A16, 47B32.

Complex symmetry of linear fractional composition operators on a half-plane

“…In Theorem 3.1, we completely characterize the cohyponormal composition operators induced by linear fractional self-maps of C + . As a consequence of this characterization we obtain a new proof of [6,Theorem 6]. For complex symmetry of composition operators on H 2 (C + ), our main results are Theorems 4.1 and 4.5, which provide concrete examples of conjugations for which linear fractional operators are complex symmetric.…”

“…(1.1) Let L(H) denote the space of all bounded linear operators on a separable complex Hilbert space H. A conjugation on H is a conjugate-linear operator satisfying C 2 = I and Cx, Cy = y, x for all x, y ∈ H. An operator T ∈ L(H) is called cohyponormal if T x ≤ T * x for all x ∈ H, normal if T T * = T * T, self-adjoint if T = T * , unitary if T T * = I = T * T and complex symmetric (or C-symmetric) if there is a conjugation C on H, for which CT * C = T. In [6], Noor and Severiano studied the composition operators on H 2 (C + ) induced by linear fractional self-maps of C + . They completely characterize the symbols that induce complex symmetric composition operators (see next theorem) and provide a new prove to characterize normal, self-adjoint and unitary composition operators on H 2 (C + ).…”

“…Noor and Severiano [6] study the complex symmetry of composition operators on H 2 (C + ). The key of this study is to analyze the cyclic behavior of the bounded composition operators induced by linear fractional self-maps of C + , which allowed them to characterize the operators that are complex symmetric.…”

Interplay between complex symmetry and Koenigs eigenfunctions

“…Recently Narayan, Sievewright, Tjani [16] characterized all non-automorphic linear factional φ for which C φ is complex symmetric. The analogous problem for the Hardy space of the half-plane H 2 (C + ) was recently solved by Noor and Severiano [18]. For more general symbols the problem still remains open.…”

Composition operators on Hardy-Smirnov spaces

“…They also characterized the non-automorphic linear self-maps of U that induce complex symmetric composition operators (see [28,Theorem 3.1]). In [29], Noor and Severiano characterized all linear fractional self-maps Φ of C + that induce complex symmetric composition operators C Φ on H 2 (C + ). 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 + ).…”