Cyclicity in Dirichlet-type spaces and extremal polynomials II: functions on the bidisk

Catherine Bénéteau, Alberto A. Condori, Constanze Liaw, Daniel Seco and Alan A. Sola

Vol. 276 (2015), No. 1, 35–58

DOI: 10.2140/pjm.2015.276.35

Abstract

We study Dirichlet-type spaces ${\mathfrak{D}}_{\alpha }$ of analytic functions in the unit bidisk and their cyclic elements. These are the functions $f$ for which there exists a sequence ${\left(p_n\right)}_{n=1}^{\infty }$ of polynomials in two variables such that $\parallel p_{n}f-1{\parallel }_{\alpha }\to 0$ as $n\to \infty$. We obtain a number of conditions that imply cyclicity, and obtain sharp estimates on the best possible rate of decay of the norms $\parallel p_{n}f-1{\parallel }_{\alpha }$, in terms of the degree of $p_n$, 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 ${\mathfrak{D}}_{\alpha }$ for $\alpha >1∕2$ (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 ($\alpha \le 1$). 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.

Keywords

cyclicity, Dirichlet-type spaces, optimal approximation, norm restrictions

Mathematical Subject Classification 2010

Primary: 32A37

Secondary: 32A36, 47A16

Milestones

Received: 15 October 2013

Revised: 10 September 2014

Accepted: 14 December 2014

Published: 1 July 2015