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arXiv 1409.3017math.FA
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狄利克雷级数的Bohr-Bergman空间上的复合算子

Composition Operators on Bohr-Bergman Spaces of Dirichlet Series

Maxime Bailleul, Ole Fredrik Brevig

备注：
Minor changes. Section 2 shortened
发表信息：
Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 1, 129--142
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摘要：

对于$α\in \mathbb{R}$，令$\mathscr{D}_α$表示由满足$\sum_{n=1}^\infty |a_n|^2/[d(n)]^α< \infty$的狄利克雷级数$f(s) = \sum_{n=1}^\infty a_n n^{-s}$组成的Hilbert空间的标度。将$\mathscr{H}^2=\mathscr{D}_0$的复合算子的Gordon-Hedenmalm定理推广到Bergman情形$α>0$ .这些复合算子由$Φ(s) = c_0 s + φ(s)$形式的函数生成，其中$c_0$是非负整数，$φ(s)$是具有一定收敛性和映射性质的狄利克雷级数。对于具有$c_0=0$的算子，发现了一个新的现象：如果$0 < α< 1$，则空间$\mathscr{D}_α$被复合算子映射到一个相同尺度的更小的空间.当$α> 1$时，空间$\mathscr{D}_α$以相同比例映射到更大的空间。此外，当$p$为奇数时，还得到了Dirichlet--Bergman空间$\mathscr{A}^p$上的复合算子关于$1 \leq p < \infty$的部分描述以及Dirichlet--Hardy空间$\mathscr{H}^p$上复合算子的新的部分结果.

英文摘要：

For $α\in \mathbb{R}$, let $\mathscr{D}_α$ denote the scale of Hilbert spaces consisting of Dirichlet series $f(s) = \sum_{n=1}^\infty a_n n^{-s}$ that satisfy $\sum_{n=1}^\infty |a_n|^2/[d(n)]^α< \infty$. The Gordon--Hedenmalm Theorem on composition operators for $\mathscr{H}^2=\mathscr{D}_0$ is extended to the Bergman case $α>0$. These composition operators are generated by functions of the form $Φ(s) = c_0 s + φ(s)$, where $c_0$ is a nonnegative integer and $φ(s)$ is a Dirichlet series with certain convergence and mapping properties. For the operators with $c_0=0$ a new phenomenon is discovered: If $0 < α< 1$, the space $\mathscr{D}_α$ is mapped by the composition operator into a smaller space in the same scale. When $α> 1$, the space $\mathscr{D}_α$ is mapped into a larger space in the same scale. Moreover, a partial description of the composition operators on the Dirichlet--Bergman spaces $\mathscr{A}^p$ for $1 \leq p < \infty$ are obtained, in addition to new partial results for composition operators on the Dirichlet--Hardy spaces $\mathscr{H}^p$ when $p$ is an odd integer.

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