On the Equivalence of the Corona and Strong Corona Problems for H ∞ $${\mathbf {H}}^\infty $$ on a Polydisk

Alexander Brudnyi ()
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Alexander Brudnyi: University of Calgary, Department of Mathematics and Statistics

Chapter 75 in Operator Theory, 2026, pp 2345-2364 from Springer

Abstract: Abstract The corona problem for the algebra of bounded holomorphic functions on a complex manifold X asks whether X is dense in the maximal ideal space of the algebra. The problem is open for such algebras on general planar domains and on n-dimensional balls and polydisks, n ≥ 2 $$n\ge 2$$ . In this chapter, we show that the corona problem on the open unit polydisk 𝔻 n ⊂ ℂ n $$\mathbb {D}^n\subset \mathbb C^n$$ is equivalent to the corona problem on the countable disjoint union of open polydisks 𝔻 n × ℕ $$\mathbb {D}^n\times \mathbb {N}$$ . It follows that the corona problem on 𝔻 n $$\mathbb {D}^n$$ is solvable if and only if the corresponding classes of Bezout equations have uniformly bounded solutions.

Keywords: Bounded holomorphic function; Maximal ideal space; Corona problem; Primary 32A38; Secondary 46J20 (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_106

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DOI: 10.1007/978-3-032-16356-1_106

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