Vol. 276, No. 1, 2015

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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
Abstract

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 a sequence (pn)n=1 of polynomials in two variables such that pnf 1α 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 1α, 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 > 12 (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.

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
Authors
Catherine Bénéteau
Department of Mathematics
University of South Florida
4202 E. Fowler Avenue
Tampa, FL 33620-5700
United States
Alberto A. Condori
Department of Mathematics
Florida Gulf Coast University
10501 FGCU Boulevard South
Fort Myers, FL 33965-6565
United States
Constanze Liaw
CASPER and Department of Mathematics
Baylor University
One Bear Place #97328
Waco, TX 76798-7328
United States
Daniel Seco
Mathematics Institute
University of Warwick
Zeeman Building
Coventry
CV4 7AL
United Kingdom
Alan A. Sola
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge
CB3 0WB
United Kingdom