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Cyclic polynomials in anisotropic Dirichlet~spaces | | Greg Knese
; Lukasz Kosinski
; Thomas J. Ransford
; Alan Sola
; | | Date: |
15 Dec 2015 | | Abstract: | Consider the Dirichlet-type space on the bidisk consisting of holomorphic
functions $f(z_1,z_2):=sum_{k,lgeq 0}a_{kl}z_1^kz_2^l$ such that
$sum_{k,lgeq 0}(k+1)^{alpha_1} (l+1)^{alpha_2}|a_{kl}|^2 <infty.$ Here the
parameters $alpha_1,alpha_2$ are arbitrary real numbers. We characterize the
polynomials that are cyclic for the shift operators on this space. More
precisely, we show that, given an irreducible polynomial $p(z_1,z_2)$ depending
on both $z_1$ and $z_2$ and having no zeros in the bidisk: if
$alpha_1+alpha_2leq 1$, then $p$ is cyclic; if $alpha_1+alpha_2>1$ and
$min{alpha_1,alpha_2}leq 1$, then $p$ is cyclic if and only if it has
finitely many zeros in the two-torus $mathbb T^2$; if
$min{alpha_1,alpha_2}>1$, then $p$ is cyclic if and only if it has no
zeros in $mathbb T^2$. | | Source: | arXiv, 1512.4871 | | Services: | Forum | Review | PDF | Favorites | |
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