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algebras generated by holomorphic and harmonic functions on the disc

Published online by Cambridge University Press:  23 September 2005

alexander j. izzo
alexander j. izzo
Affiliation:
department of mathematics and statistics, bowling green state university, bowling green, oh 43403, usaaizzo@math.bgsu.edu current address: department of mathematics, brown university, box 1917, providence, ri 02912, usa
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Abstract

let $e$ be a subset of the boundary of the open unit disc $d$, and let $a$ be the algebra of bounded holomorphic functions on $d$ that extend continuously to $d \cup e$. it is shown that if $f$ is a bounded harmonic function on $d$ that extends continuously to $d \cup e$ and is not holomorphic, then the uniformly closed algebra $a[f]$ generated by $a$ and $f$ contains $c({\overline{d}})$. this result contains as special cases a result on the disc algebra due to čirka and a result on $h^{\infty}(d)$ due to axler and shields. a stronger form of the result, in which $f$ is allowed to have discontinuities on a small subset of $e$, is also established.

Keywords

46j1046j1530h05
Type
papers
Information
Bulletin of the London Mathematical Society , Volume 37 , Issue 5 , October 2005 , pp. 761 - 770
Copyright
the london mathematical society 2005

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Footnotes

this paper was presented to the american mathematical society in preliminary form, on october 13, 2002, under the title interpolating between a theorem of čirka and a theorem of axler and shields.