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2020-04

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cc-by (c) Massaneda Clares, Francesc Xavier et al., 2020
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/192522

From $H^\infty$ to $N$. Pointwise properties and algebraic structure in the Nevanlinna class

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https://doi.org/10.1515/conop-2020-0007

Abstract

This survey shows how, for the Nevanlinna class $\mathcal{N}$ 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 $\mathcal{H}^{\infty}$ : 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 $\mathcal{H}^{\infty}$ can be transposed to $\mathcal{N}$ 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.

Subject

Teoria de Nevanlinna, Funcions de variables complexes

Subject (English)

Nevanlinna theory, Functions of complex variables

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Articles publicats en revistes (Matemàtiques i Informàtica)

Citation

MASSANEDA CLARES, Francesc Xavier and THOMAS, Pascal J. From $H^\infty$ to $N$. Pointwise properties and algebraic structure in the Nevanlinna class. Concrete Operators. 2020. Vol. 7, num. 1, pags. 91-115. ISSN 2299-3282. [consulted: 18 of August of 2026]. Available at: https://hdl.handle.net/2445/192522

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