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On the composition of Frostman Blaschke products

100%
Akeroyd J., Gorkin P.
Studia Mathematica
|
2013
|
tom 219
|
nr 2
177-191
EN
We construct an infinite uniform Frostman Blaschke product B such that B ∘ B is also a uniform Frostman Blaschke product. We also show that the set of uniform Frostman Blaschke products is open in the set of inner functions with the uniform norm.
2
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Pervasive algebras and maximal subalgebras

100%
Gorkin P., O'Farrell A.
Studia Mathematica
|
2011
|
tom 206
|
nr 1
1-24
EN
A uniform algebra A on its Shilov boundary X is maximal if A is not C(X) and no uniform algebra is strictly contained between A and C(X). It is essentially pervasive if A is dense in C(F) whenever F is a proper closed subset of the essential set of A. If A is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show: (1) If A is pervasive and proper, and has a nonconstant unimodular element, then A contains an infinite descending chain of pervasive subalgebras on X. (2) It is possible to find a compact Hausdorff space X such that there is an isomorphic copy of the lattice of all subsets of ℕ in the family of pervasive subalgebras of C(X). (3) In the other direction, if A is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word "strongly" is removed. We discuss examples involving Dirichlet algebras, A(U) algebras, Douglas algebras, and subalgebras of $H^{∞}(𝔻)$, and develop new results that relate pervasiveness, maximality, and relative maximality to support sets of representing measures.
3
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An upper bound for the distance to finitely generated ideals in Douglas algebras

81%
Gorkin P., Mortini R., Suárez D.
Studia Mathematica
|
2001
|
tom 148
|
nr 1
23-36
EN
Let f be a function in the Douglas algebra A and let I be a finitely generated ideal in A. We give an estimate for the distance from f to I that allows us to generalize a result obtained by Bourgain for $H^{∞}$ to arbitrary Douglas algebras.
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