Contractive zero-divisors in Bergman spaces

Abstract

Generalizing a recent result of H. Hedenmalm for p = 2, a contractive zero-divisor is found in the Bergman space Ap over the unit disk for 1 < p < oo. This is a function G £ Ap with \\G\\P = 1 and a prescribed zero-set {ζj} , uniquely determined by the contractive property \\f/G\\p < \\f\\p for all f e Ap which vanish on {ζj} . 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. 0. Introduction. It is well known that the zero-sets {ζj} of functions / in the Hardy space Hp are characterized by the Blaschke condition ]Γ}(1 — \ζj\) < oo, which guarantees the convergence of the Blaschke product H,ζj), where *(z,O = ψy ^ if ζ Φ 0, and b(z, 0) = z. A basic theorem of F. Riesz (see [3], Ch. 2) asserts that f/B is a nonvanishing function in Hp with norm equal to that of /. This simple fact plays an important role in the theory of Hp spaces, and it would be desirable to find an analogue for the Bergman spaces. We shall see in this paper that while isometric zerodivisors are not available in the Bergman spaces, there is an essentially unique contractive divisor of unit norm associated with every zero-set. A function / analytic in the unit disk D is said to belong to the Bergman space Ap , where 0 < p < oo, if