Integral Equations and Operator Theory

, Volume 86, Issue 4, pp 453–473 | Cite as

Weak Product Spaces of Dirichlet Series

  • Ole Fredrik Brevig
  • Karl-Mikael PerfektEmail author


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.


Dirichlet series Hankel form Square function Weak product space 

Mathematics Subject Classification

Primary 47B35 Secondary 30B50 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and Technology (NTNU)TrondheimNorway
  2. 2.Centre for Mathematical SciencesLund UniversityLundSweden

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