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

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 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.