Complex Symmetry of Composition Operators on Hilbert Spaces of Entire Dirichlet Series

Abstract

A criterion for boundedness of composition operators acting on a class of Hilbert spaces of entire Dirichlet series, namely the class H(E, βS), was obtained in J. Math. Anal. Appl. ( 2013) for those spaces that do not contain non-zero constant functions, while other possibilities were not studied. In this paper, we first provide a complete characterization of boundedness of composition operators on any space H(E, βS), which may or may not contain constant functions. We then study complex symmetry of composition operators on H(E, βS), via analysis of composition conjugations.

Weighted holomorphic Dirichlet series and composition operators with polynomial symbols

“…In this subsection, we obtain an important necessary condition which a general symbol ϕ inducing a bounded composition operator C ϕ must satisfy. See [8,19] for analogous results in the entire case. First, note that in the case when β * = ±∞, all f ∈ H(β, Λ) converge on the proper right half-plane C L 2 −β * and in particular, satisfy D f = ∞ (by Valiron's formulae).…”

“…In [8], a characterization for boundedness of C ϕ on H(β, Λ) in the case of β * = ∞ was proven. Specifically, the authors prove the following analogue to Theorem 3.14:…”

“…• Conjugations: Theorem 6.3 (cf. [9]). It is worth noting that due to Theorem 7.1, we can weaken the assumption on ξ to simply assuming ξ is entire.…”

“…Note that for the case β * = ∞, we have entire Dirichlet series which have been studied quite well (see, e.g., [8] and related references). Therefore, in the sequel we assume that the condition β * = ±∞ always holds.…”

“…The case of general Dirichlet series is less studied. Nevertheless, when X is some weighted space of entire Dirichlet series of type Λ satisfying a certain property, properties of composition operators acting on X are quite well understood (see [9,18,19]). In [8], using Liouville's theorem, it is proved that if C ϕ defines a bounded composition operator on a weighted Hilbert space of entire Dirichlet series, then ϕ must be an affine function.…”