Abstract
Let H2 denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators Cφ on H2 which are generated by symbols of the form φ(s)=c0s+∑n≥1cnn−s, in the case that c0≥1. If only a subset P of prime numbers features in the Dirichlet series of φ, then the operator Cφ admits an associated orthogonal decomposition. Under sparseness assumptions on P we use this to asymptotically estimate the approximation numbers of Cφ. Furthermore, in the case that φ is supported on a single prime number, we affirmatively settle the problem of describing the compactness of Cφ in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps.
Highlights
Let H 2 be the Hilbert space of Dirichlet series f (s) = n≥1 bnn−s with squaresummable coefficients
Gordon and Hedenmalm [10] established that the composition operator Cφf = f ◦ φ defines a bounded composition operator on H 2 if and only if φ belongs to the Gordon–Hedenmalm class G
The Gordon–Hedenmalm class G consists of the analytic functions φ : C1/2 → C1/2 of the form φ(s) = c0s + cnn−s = c0s + φ0(s), n=1 where c0 is a non-negative integer and the Dirichlet series φ0 converges uniformly in Cε for every ε > 0 and satisfies the following mapping properties: (a) If c0 = 0, φ0(C0) ⊆ C1/2. (b) If c0 ≥ 1, either φ0(C0) ⊆ C0 or φ0 ≡ iτ for some τ ∈ R
Summary
To exemplify the type of estimates which can be obtained from Theorem 1.3, let P be a set of prime numbers and consider the affine symbol (1.3). Using Theorem 1.3, we shall obtain the following estimate for the approximation numbers of composition operators generated by affine symbols φ ∈ G≥1. In Theorem 4.2 we shall consider some examples of affine symbols supported on infinite but very sparse sets of prime numbers. To the case of affine maps discussed above, Theorem 1.3 (a) does not provide the correct lower bound when θ = 0 In this case we shall instead proceed via the change of variable formula of Lemma 6.2 and detailed analysis of the restricted counting function. In the classical setting of H2(D), detailed studies of the approximation numbers of composition operators generated by symbols that map into an angle are carried out in [12] and [16]. The notation ≫ indicates the reverse estimate, and f (x) ≍ g(x) means that f (x) ≪ g(x) and g(x) ≪ f (x)
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