From H∞ to N. Pointwise Properties and Algebraic Structure in The
Concr. Oper. 2020; 7:91–115 Review Article Open Access Xavier Massaneda and Pascal J. Thomas From H∞ to N. Pointwise properties and algebraic structure in the Nevanlinna class https://doi.org/10.1515/conop-2020-0007 Received November 4, 2019; accepted March 20, 2020 Abstract: This survey shows how, for the Nevanlinna class N of the unit disc, one can dene and often charac- terize 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 briey discuss the situation for the related Smirnov class. Keywords: Nevanlinna class, interpolating sequences, Corona theorem, sampling sets, Smirnov class MSC: 30H15 , 30H05, 30H80 1 Introduction 1.1 The Nevanlinna class ∞ 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∞. The goal of this paper is to survey those results.
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