Contractive zero-divisors in Bergman spaces

Peter Larkin Duren, Dmitry Khavinson, Harold Seymour Shapiro and Carl Sundberg

Vol. 157 (1993), No. 1, 37–56

DOI: 10.2140/pjm.1993.157.37

Abstract

Generalizing a recent result of H. Hedenmalm for p = 2, a contractive zero-divisor is found in the Bergman space A^p over the unit disk for 1 ≤ p < ∞. This is a function G ∈ A^p with ∥G∥_p = 1 and a prescribed zero-set {ζ_j}, uniquely determined by the contractive property ∥f∕G∥_p ≤∥f∥_p for all f ∈ A^p which vanish on {ζ_j}. The proof uses the positivity of the biharmonic Green function of the disk. For a finite zero-set, the canonical divisor G is represented explicitly in terms of the Bergman kernel of a certain weighted A² space. It is then shown that G has an analytic continuation to a larger disk.

Mathematical Subject Classification 2000

Primary: 30D99

Secondary: 30D55, 30H05, 46E15

Milestones

Received: 2 July 1991

Published: 1 January 1993