Weak Product Spaces of Dirichlet Series

Abstract

Let \({\mathscr{H}^2}\) denote the space of ordinary Dirichlet series with square summable coefficients, and let \({\mathscr{H}^{2}_{0}}\) denote its subspace consisting of series vanishing at \({+\infty}\). We investigate the weak product spaces \({\mathscr{H}^{2} \odot \mathscr{H}^{2}}\) and \({\mathscr{H}^{2}_{0} \odot \mathscr{H}^{2}_{0}}\), finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that \({\mathscr{H}^{2}_{0} \odot \mathscr{H}^{2}_{0}}\) does not contain the infinite-dimensional subspace of \({\mathscr{H}^2}\) of series which lift to linear functions on the infinite polydisc. The problems considered stem from questions about the dual spaces of these weak product spaces, and are therefore naturally phrased in terms of multiplicative Hankel forms. We show that there are bounded, even Schatten class, multiplicative Hankel forms on \({\mathscr{H}^{2}_{0} \times \mathscr{H}^{2}_{0}}\) whose analytic symbols are not in \({\mathscr{H}^2}\). Based on this result we examine Nehari’s theorem for such Hankel forms. We define also the skew product spaces associated with \({\mathscr{H}^{2} \odot \mathscr{H}^2}\) and \({\mathscr{H}^{2}_{0} \odot \mathscr{H}^{2}_{0}}\), with respect to both half-plane and polydisc differentiation, the latter arising from Bohr’s point of view. In the process we supply square function characterizations of the Hardy spaces \({\mathscr{H}^p}\), for \({0 < p < \infty}\), from the viewpoints of both types of differentiation. Finally we compare the skew product spaces to the weak product spaces, leading naturally to an interesting Schur multiplier problem.

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Correspondence to Karl-Mikael Perfekt.

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O. F. Brevig is supported by Grant 227768 of the Research Council of Norway. K.-M. Perfekt was supported by the Swedish Research Council under contract 621-2014-5159 to Johan Helsing.

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Brevig, O.F., Perfekt, K. Weak Product Spaces of Dirichlet Series. Integr. Equ. Oper. Theory 86, 453–473 (2016). https://doi.org/10.1007/s00020-016-2320-3

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Mathematics Subject Classification

  • Primary 47B35
  • Secondary 30B50

Keywords

  • Dirichlet series
  • Hankel form
  • Square function
  • Weak product space