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- Aleman, Alexandru, et al.
(author)
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CYCLICITY IN THE DRURY-ARVESON SPACE AND OTHER WEIGHTED BESOV SPACES
- 2024
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In: Transactions of the American Mathematical Society. - 0002-9947. ; 377:2, s. 1273-1298
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Journal article (peer-reviewed)abstract
- Let H be a space of analytic functions on the unit ball Bd in Cd with multiplier algebra Mult(H). A function f ∈ H is called cyclic if the set [f], the closure of {ϕf : ϕ ∈ Mult(H)}, equals H. For multipliers we also consider a weakened form of the cyclicity concept. Namely for n ∈ N0 we consider the classes Cn(H) = {ϕ ∈ Mult(H): ϕ /= 0, [ϕn] = [ϕn+1]}. Many of our results hold for N:th order radially weighted Besov spaces on Bd, H = BωN, but we describe our results only for the Drury-Arveson space Hd2 here. Letting Cstable[z] denote the stable polynomials for Bd, i.e. the d-variable complex polynomials without zeros in Bd, we show that if d is odd, then Cstable[z] ⊆ Cd−1 (Hd2), and 2 if d is even, then Cstable[z] ⊆ Cd2 −1(Hd2). For d = 2 and d = 4 these inclusions are the best possible, but in general we can only show that if 0 ≤ n ≤ d4 − 1, then Cstable[z] Cn(Hd2). For functions other than polynomials we show that if f, g ∈ Hd2 such that f/g ∈ H∞ and f is cyclic, then g is cyclic. We use this to prove that if f, g extend to be analytic in a neighborhood of Bd, have no zeros in Bd, and the same zero sets on the boundary, then f is cyclic in ∈ Hd2 if and only if g is. Furthermore, if the boundary zero set of f ∈ Hd2 ∩ C(Bd) embeds a cube of real dimension ≥ 3, then f is not cyclic in the Drury-Arveson space.
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