Weak products of complete Pick spaces
Alexandru Aleman, Michael Hartz, John E. McCarthy, Stefan Richter

TL;DR
This paper studies the weak product of certain function spaces on the unit ball, showing they are contained in the Smirnov class and establishing a correspondence between their multiplier invariant subspaces, with implications for interpolation theory.
Contribution
It demonstrates that weak products of complete Pick spaces are contained in the Smirnov class and establishes a bijection between their multiplier invariant subspaces, extending to many weighted Besov spaces.
Findings
Weak product functions are quotients of multipliers with cyclic denominators.
A bijection exists between multiplier invariant subspaces of the space and its weak product.
Provides an alternative proof for interpolating sequences in the Drury-Arveson space.
Abstract
Let be the Drury-Arveson or Dirichlet space of the unit ball of . The weak product of is the collection of all functions that can be written as , where . We show that is contained in the Smirnov class of , i.e. every function in is a quotient of two multipliers of , where the function in the denominator can be chosen to be cyclic in . As a consequence we show that the map establishes a 1-1 and onto correspondence between the multiplier invariant subspaces of and of . The results hold for many weighted Besov spaces in the unit ball of $\mathbb…
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