Composition operators on Bohr-Bergman spaces of Dirichlet series (Q2790812)

For \(\alpha \in \mathbb{R}\), the scale of Hilbert spaces of Dirichlet series \({\mathcal{D}}_{\alpha}\) is defined as NEWLINE\[NEWLINE {\mathcal{D}}_{\alpha} = \left\{ f : s \rightarrow f(s)= \sum\limits_{n=1}^{\infty}a_nn^{-s} \mid \| f\| _{ {\mathcal{D}}_{\alpha}} = \left(\sum\limits_{n=1}^{\infty}\frac{|a|^2}{[d(n)]^2}\right)^{\frac{1}{2}} < \infty \right\},NEWLINE\]NEWLINE where \(d(n)\) denotes the number of divisors of the integer \(n\). The Hardy space is the case \(\alpha=0\). Let \({\mathcal{A}}^2\) denote \({\mathcal{D}}_{1}\), which is the a Dirichlet series analogue to the classical unweighted Bergman space of the unit disc, \(A^2(\mathbb{D})\). For \(\theta \in \mathbb{R}\), let \(C_{\theta}\) denote the half-plane \(\{ \sigma+it : t \in \mathbb{R} \;\, \sigma > \theta \}\). The Gordon-Hedenmalm class, denoted by \(\mathcal{G}\), is the set of functions \( \phi : C_{1/2} \rightarrow C_{1/2}\) of the form \(\phi(s)=c_0s+\sum\limits_{n=1}^{\infty}a_nn^{-s}\), where \(c_0\) is a non-negative integer called the characteristic of \(\phi\). It is known that the Dirichlet series \(\varphi=\sum\limits_{n=1}^{\infty}a_nn^{-s}\) converges uniformly in \(C_{\varepsilon}\) (\(\varepsilon > 0 \)) and has the following mapping properties: (a) If \(c_0 = 0\), then \(\varphi(C_0) \subset C_{1/2}\). (b) If \(c _0 \geq 1\), then either \(\varphi = 0\) or \(\varphi(C_0) \subset C_0\). A slightly stronger version of the Gordon-Hedenmalm Theorem states that \( \phi : C_{1/2} \rightarrow C_{1/2}\) induces a composition operator on \({\mathcal{H}}^2\) if and only if \(\phi \in \mathcal{G}\). The paper under review studies extension of the Gordon-Hedenmalm Theorem on composition operators for \({\mathcal{H}}^2 ={\mathcal{D}}_0\) to \({\mathcal{D}}_{\alpha}\) for \(\alpha > 0\). For \(\alpha > 0\), the authors prove that \( \phi : C_{1/2} \rightarrow C_{1/2}\) induces a composition operator \(C_{\phi} : {\mathcal{D}}_{\alpha} \rightarrow {\mathcal{D}}_{\beta}\), where NEWLINE\[NEWLINE\beta = \begin{cases} 2^{\alpha}-1, & c_0=0, \\ \alpha, & c_0 \geq 1, \end{cases} NEWLINE\]NEWLINE if and only if \(\phi \in {\mathcal{G}}\). Morever, if \(c_0 \geq 1\), then the operator is a contraction. For a Dirichlet series \(f : s \mapsto f(s)= \sum\limits_{n=1}^{\infty}a_nn^{-s}\), the Bohr lift of \(f\), denoted by \({\mathcal{B}}f\), on the polydisk \(\mathbb{D}^{\infty} = \{ z=(z_1,\dots z_n,\dots) : |z_j| < 1 \}\), is defined as the power series NEWLINE\[NEWLINE({\mathcal{B}}f)(z) = \sum\limits_{n=1}^{\infty}a_nz^{k(n)},NEWLINE\]NEWLINE where \(k(n)=(k_1,k_2,\dots,k_n,\dots)\) and \(n=\prod_jp_j^{k_j}\). The space \({\mathcal{H}}^2\) is identified with the Hardy space \(H^2(\mathbb{T}^{\infty})\), where \(\mathbb{T}^{\infty}\) is the distinguished boundary of \(\mathbb{D}^{\infty}\) under the Bohr lift. A similar identification for the spaces \({\mathcal{H}}^p\) and the corresponding extension of the Gordon-Hedenmalm Theorem were obtained in [\textit{F. Bayart}, Monatsh. Math. 136, No. 3, 203--236 (2002; Zbl 1076.46017)] with one possible exception, namely, the sufficiency condition (a) is proved only for the case where \(p\) is an even integer. In the present paper, the authors present partial results on the description of composition operators on \({\mathcal{H}}^p\) for the case where \(c_0=0\) and \(p\) is odd. For \(1 \leq p < \infty\), the Bohr-Bergmann space \({\mathcal{A}}^p\) is defined as NEWLINE\[NEWLINE{\mathcal{A}}^p = \left\{ f : s \mapsto f(s)= \sum\limits_{n=1}^{\infty}a_nn^{-s} \mid \| f\| _{{\mathcal{A}}^p} = \left(\int\limits_{\mathbb{D}^{\infty}}|({\mathcal{B}}f)(z)|^pd\nu(z)\right)^{1/p} <\infty \right\}.NEWLINE\]NEWLINE For \(1 \leq p < \infty\), the authors also give a partial description of composition operators on \({\mathcal{A}}^p\). The paper concludes by giving a complete description of composition operators on a certain class of Bergmann-type spaces introduced by \textit{J. E. McCarthy} [Trans. Am. Math. Soc. 356, No. 3, 881--893 (2004; Zbl 1039.30001)].