The Corona Problem for Kernel Multiplier Algebras

详细信息    查看全文

• 作者：Eric T. Sawyer ; Brett D. Wick
• 刊名：Integral Equations and Operator Theory
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：86
• 期：4
• 页码：495-544
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1420-8989
• 卷排序：86

文摘

We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in \(\mathbb {C} ^{n}\), and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions \(k_{x}\left( y\right) \) of certain Hilbert function spaces \( \mathcal {H}\) are assumed to be invertible multipliers on \(\mathcal {H}\), and then we continue a research thread begun by Agler and McCarthy in 1999, and continued by Amar in 2003, and most recently by Trent and Wick in 2009. In dimension \(n=1\) we prove the corona theorem for the kernel multiplier algebras of Besov-Sobolev Banach spaces in the unit disk, extending the result for Hilbert spaces \(H^\infty \cap Q_p\) by Nicolau and Xiao.