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The generalized corona theorem. (English) Zbl 0834.46034
This is another beautiful paper on the ideal theory of $$H^\infty$$ by one of the leading experts (R.M.) and colleagues. $$H^\infty$$ is the ring of bounded holomorphic functions in the open unit disk $$\mathbb{D}$$. For $$f_1,\dots, f_N\in H^\infty$$, $$J:= \{f\in H^\infty: |f|\leq C_f \sum^N_{j= 1} |f_j|$$, for some finite constant $$C_f\}$$ is an ideal that contains the ideal $$I$$ generated by $$f_1,\dots, f_N$$, and simple examples show that proper containment can occur. L. Carleson’s famous Corona Theorem asserts that equality occurs if $$J= H^\infty$$. It is proved that if $$N= 2$$ and $$f_1$$, $$f_2$$ have no common non-invertible $$H^\infty$$-factor, then $$I= J$$ if and only if either $$I$$ or $$J$$ contains an interpolating Blaschke product. [Earlier work of V. Tolokonnikov is relevant, especially J. Soc. Math. 27, 2549-2553 (1984; Zbl 0546.46046).] But for $$N= 3$$ an (easy) example is constructed of $$I= J$$ without this ideal containing an interpolating Blaschke product. Denote by $$M$$ the maximal ideal space of $$H^\infty$$, and by $$\widehat f: M\to \mathbb{C}$$ the Gelfand transform of $$f\in H^\infty$$. The order of $$m\in M$$ as a zero of $$\widehat f$$ is defined in a natural way and the minimum of these over $$f\in I$$, denoted $$\text{ord}(I, m)$$, plays a significant role. Use is made of an earlier result of the second author’s [Analysis 14, No. 1, 67-73 (1984; Zbl 0810.46055)]that an ideal $$I\subsetneqq H^\infty$$ is generated by interpolating Blaschke products if and only if $$\text{ord}(I, m)\in \{0, 1\}$$ for every $$m\in M$$. Thomas Wolff showed some time ago that $$f\in J$$ implies $$f^3\in I$$ always holds, and he posed the problem [cf. Lect. Notes Math. 1043 (1984; Zbl 0545.30038)] whether the stronger conclusion $$f^2\in I$$ always holds as well.
In the second part of this paper, the authors provide an affirmative answer under the additional assumption that at every $$m\in M$$, $$\text{ord}(\widehat f_j, m)$$ is finite for at least one $$j\in \{1,\dots, N\}$$; this hypothesis is (nontrivially) equivalent to the $$\widehat f_j$$ not all vanishing at any one-point Gleason part in $$M$$. The methods used are the standard ones developed by Wolff for proving the Corona theorem and involve Carleson measures, CN (= Carleson-Newman)- sequences and CN-Blaschke products.

##### MSC:
 46J20 Ideals, maximal ideals, boundaries 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 30D55 $$H^p$$-classes (MSC2000) 30D50 Blaschke products, etc. (MSC2000)
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