Strong algebrability and residuality on certain sets of analytic functions.(English)Zbl 1451.46022

In this interesting paper the authors investigate the presence of algebraic structures within certain sets of analytic functions. For instance, they show that the set of analytic functions from $$\mathbb{C}^{2}$$ into $$\mathbb{C}^{2}$$ which are not Lorch-analytic is spaceable and strongly $$\mathfrak{c}$$-algebrable (where $$\mathfrak{c}$$ denotes the cardinality of the continuum) in the space of entire functions from $$\mathbb{C}^{2}$$ into $$\mathbb{C}^{2}$$. They also show that this set is not residual. Similar properties are investigated for the set of functions which belong to the the disk algebra but not a Dales-Davie algebra. Some properties of lineability are also obtained.

MSC:

 46B87 Lineability in functional analysis 46G20 Infinite-dimensional holomorphy 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 46J10 Banach algebras of continuous functions, function algebras 46T25 Holomorphic maps in nonlinear functional analysis
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