RT Journal Article
T1 Cyclicity in Dirichlet-type spaces and extremal polynomials II: functions on the bidisk
A1 Beneteau, Catherine
A1 Condori, Alberto
A1 Liaw, Constanze
A1 Seco Forsnacke, Daniel
A1 Sola, Alan A.
AB We study Dirichlet-type spaces Dα of analytic functions in the unit bidisk and their cyclic elements. These are the functions f for which there exists asequence(pn)∞n=1 of polynomials in two variables such that ‖pnf−1‖α→0 as n→∞. We obtain a number of conditions that imply cyclicity, and obtain sharp estimates on the best possible rate of decay of the norms ‖pnf−1‖α,in terms of the degree of pn, for certain classes of functions using results concerning Hilbert spaces of functions of one complex variable and comparisons between norms in one and two variables. We give examples of polynomials with no zeros on the bidisk that are not cyclic in Dα for α >1/2 (including the Dirichlet space); this is in contrast with the one-variable case where all nonvanishing polynomials are cyclic in Dirichlet-type spaces that are not algebras (α≤1). Further, we point out the necessity of a capacity zero condition on zero sets (in an appropriate sense) for cyclicity in the setting of the bidisk, and conclude by stating some open problems.
PB Mathematical Sciences Publishers (MSP)
SN 0030-8730
YR 2015
FD 2015-07
LK https://hdl.handle.net/10016/32841
UL https://hdl.handle.net/10016/32841
LA eng
NO Liaw is partially supported by the NSF grant DMS-1261687. Seco is supported by ERC Grant 2011-ADG-20110209 from EU programme FP2007-2013, and by MEC/MICINN Project MTM2011-24606. Sola acknowledges support from the EPSRC under grant EP/103372X/1.
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RD 27 jul. 2024