## Abstract

Let H be the Drury-Arveson or Dirichlet space of the unit ball of C^{d}. The weak product H ☉ H of H is the collection of all functions h that can be written as h =∑^{∞}_{n=}_{1} f_{n}g_{n}, where ∑^{∞}_{n}=_{1} ||f_{n}|| ||g_{n}|| < ∞. We show that H ☉ H is contained in the Smirnov class of H; that is, every function in H ☉ H is a quotient of two multipliers of H, where the function in the denominator can be chosen to be cyclic in H . As a consequence, we show that the map N → clos_{H ☉H} N establishes a one-to-one and onto correspondence between the multiplier invariant subspaces of H and of H ☉ H . The results hold for many weighted Besov spaces H in the unit ball of C^{d} provided the reproducing kernel has the complete Pick property. One of our main technical lemmas states that, for weighted Besov spaces H that satisfy what we call the multiplier inclusion condition, any bounded column multiplication operator H → ⊕^{∞}_{n=}_{1} H induces a bounded row multiplication operator ⊕^{∞}_{n=}_{1} H → H . For the Drury-Arveson space H_{d}^{2} this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions.

Original language | English |
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Pages (from-to) | 325-352 |

Number of pages | 28 |

Journal | Indiana University Mathematics Journal |

Volume | 70 |

Issue number | 1 |

DOIs | |

State | Published - 2021 |

## Keywords

- Besov space
- Complete Pick space
- Multiplier
- Smirnov class