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arXiv 1401.3286math.FAmath.OA
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狄利克雷级数在希尔伯特空间上的乘子

Multipliers on Hilbert Spaces of Dirichlet Series

Eric Stetler

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摘要：

本文研究了一类具有加权范数的狄利克雷级数的希尔伯特空间及其相应的乘子代数。对于正数序列$\{w_n\}_{n=n_0}^\infty $，定义\[\mathcal H^\textbf{w}=\left\{\sum_{n=n_0}^\infty a_nn^{-s}:\sum_{n=n_0}^\infty |a_n|^2 w_n<\infty\right\}.\] Hedenmalm，Lindqvist和Seip考虑了$w_n\equiv 1$的情况，并将该空间的$\mathcal H^\textbf{w}$的乘子代数分类为\cite {HLS}。在\cite{M}中，McCarthy对$\mathcal H^\textbf{w}$上的乘数进行了分类，当权重由\[w_n=\int_0^\infty n^{-2σ}dμ(σ),\]给出时，其中$μ$是正Radon测度，支持$\{0\}$，$n_0$是该积分有限的最小正整数。假设权重是乘法的，而不是由度量给出的，将得出类似的结果。特别是，上和下界的乘子的运营商的范数将获得，在某些半平面上，从这些权重产生的希尔伯特空间的值。最后，一些数论的重量序列将探讨和相应的希尔伯特空间的乘子代数确定的等距同构，提供的例子，麦卡锡的结论成立，但在其他假设的重量。

英文摘要：

In this paper, certain classes of Hilbert spaces of Dirichlet series with weighted norms and their corresponding multiplier algebras will be explored. For a sequence $\{w_n\}_{n=n_0}^\infty $ of positive numbers, define \[\mathcal H^\textbf{w}=\left\{\sum_{n=n_0}^\infty a_nn^{-s}:\sum_{n=n_0}^\infty |a_n|^2 w_n<\infty\right\}.\] Hedenmalm, Lindqvist and Seip considered the case in which $w_n\equiv 1$ and classified the multiplier algebra of $\mathcal H^\textbf{w}$ for this space in \cite {HLS}. In \cite{M}, McCarthy classified the multipliers on $\mathcal H^\textbf{w}$ when the weights are given by \[w_n=\int_0^\infty n^{-2σ}dμ(σ),\] where $μ$ is a positive Radon measure with $\{0\}$ in its support and $n_0$ is the smallest positive integer for which this integral is finite. Similar results will be derived assuming the weights are multiplicative, rather than given by a measure. In particular, upper and lower bounds on the operator norms of the multipliers will be obtained, in terms of their values on certain half planes, on the Hilbert spaces resulting from these weights. Finally, some number theoretic weight sequences will be explored and the multiplier algebras of the corresponding Hilbert spaces determined up to isometric isomorphism, providing examples where the conclusion of McCarthy's result holds, but under alternate hypotheses on the weights.

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