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Bounded Point Derivations on Certain Function Algebras

  • J. E. Brennan
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 261)

Abstract

Let X be a compact nowhere dense subset of the complex plane ℂ, let C(X) be the linear space of all continuous functions on X endowed with the uniform norm, and let dA denote two-dimensional Lebesgue (or area) measure in ℂ. Denote by R(X) the closure in C(X) of the set of all rational functions having no poles on X. It is well known that if X is sufficiently massive, then the functions in R(X) can inherit many of the properties usually associated with the analytic functions, such as unlimited degrees of differentiability and even the uniqueness property itself. Here we shall examine the extent to which some of those properties are inherited by the larger algebra H ∞ (X), which by definition is the weak-* closure of R(X) in L(X) = L(X, dA).

Keywords

Point derivation monogeneity Swiss cheese peak point analytic capacity Wang’s theorem 

Mathematics Subject Classification (2010)

Primary 30H50 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA

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