On the Brown–Shields conjecture for cyclicity in the Dirichlet space期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

On the Brown–Shields conjecture for cyclicity in the Dirichlet space

Authors:	Omar El-Fallah Karim Kellay Thomas Ransford

Affiliation:	^aDépartement de Mathématiques, Université Mohamed V, B.P. 1014 Rabat, Morocco;^bCMI, LATP, Université de Provence, 39, rue F. Joliot-Curie, 13453 Marseille cedex 13, France;^cDépartement de Mathématiques et de Statistique, Université Laval, Québec (QC), Canada G1V 0A6

Abstract:	Let be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function to be cyclic, i.e. for to be dense in . This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in iff it is outer and its zero set (defined appropriately) is of capacity zero.

Keywords:	Dirichlet integral Dirichlet space Invariant subspace Cyclic function Outer function Logarithmic capacity Brown– Shields conjecture Bergman– Smirnov exceptional set
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