Abstract
Abstract This survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.
Highlights
This survey shows how, for the Nevanlinna class N of the unit disc, one can de ne and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions H∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras
For H∞, and for a function algebra in general, under the heading “pointwise properties” we mean the characterization of zero sets, interpolating sequences, and sets of determination; while the aspects of algebraic structure of H∞ we are interested in concern its ideals: the Corona theorem, which says that D is dense in the maximal ideal space of H∞, the computation of stable rank, and the determination of invertibility in a quotient algebra from the values of an equivalence class over the set where they coincide
The analogue is true for N, but unlike in the bounded case, this is equivalent to the inner function being a Blaschke product of a nite number of Nevanlinna interpolating sequences
Summary
The class H∞ of bounded holomorphic functions on the unit D disc enjoys a wealth of analytic and algebraic properties, which have been explored for a long time with no end in sight, see e.g. [11], [38]. The class H∞ of bounded holomorphic functions on the unit D disc enjoys a wealth of analytic and algebraic properties, which have been explored for a long time with no end in sight, see e.g. [11], [38] These last few years, some similar properties have been explored for the Nevanlinna class, a much larger algebra which is in some respects a natural extension of H∞. According the most common de nition, the Nevanlinna class is the algebra of analytic functions π.
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