Finite unions of interpolating sequences for Hardy spaces

TL;DR

This paper studies finite unions of interpolating sequences for Hardy spaces, exploring their properties and characterizations, and introduces new insights and simpler proofs related to their role in Bergman spaces.

Contribution

It provides new characterizations of finite unions of interpolating sequences for Hardy spaces and offers a simplified proof of a known characterization.

Findings

Finite unions of interpolating sequences are characterized as interpolating for a broader class.

Blaschke products with such zero sequences have operators with closed range on Bergman spaces.

New and simpler proofs of existing characterizations are presented.

Abstract

A sequence which is a finite union of interpolating sequences for $H^\infty$ have turned out to be especially important in the study of Bergman spaces. The Blaschke products $B(z)$ with such zero sequences have been shown to be exactly those such that the multiplication $f \mapsto fB$ defines an operator with closed range on the Bergman space. Similarly, they are exactly those Blaschke products that boundedly divide functions in the Bergman space which vanish on their zero sequence. There are several characterizations of these sequences, and here we add two more to those already known. We also provide a particularly simple new proof of one of the known characterizations. One of the new characterizations is that they are interpolating sequences for a more general interpolation problem.