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Reduced Cowen sets. (English) Zbl 0990.47027
Let $$\mathbb{D}$$ denote the unit disk of the complex plane and $$\partial\mathbb{D}$$ its boundary. Let $${\mathcal H}^2$$ be the Hardy space, that is, the space of those functions $$f(z)= \sum_{n=0}^\infty a_n z^n$$ analytic on $$\mathbb{D}$$ for which the norm $\|f\|^2_2= \sum^\infty_{n=0} |a_n|^2$ is finite. The Hardy space is considered as a subspace of $$L^2(\partial\mathbb{D})$$. Given a function $$\varphi\in L^\infty(\partial\mathbb{D})$$, the Toeplitz operator with symbol $$\varphi$$ is the operator $$T_\varphi$$ on the Hardy space $${\mathcal H}^2$$ defined by $$T_\varphi f= P(\varphi f)$$, where $$P$$ denotes the orthogonal projection from $$L^2(\partial\mathbb{D})$$ onto $${\mathcal H}^2$$. Recall that an operator $$T$$ is said to be hyponormal if $$T^* T-TT^*\geq 0$$. In particular, subnormal operators are hyponormal. The problem of determining which symbols induce hyponormal Toeplitz operators was completely solved by C. C. Cowen [Proc. Am. Math. Soc. 103, No. 3, 809-812 (1988; Zbl 0668.47021)]. Let $${\mathcal H}^\infty$$ denote the space of bounded analytic functions on $$\mathbb{D}$$. Let $$\varphi\in L^\infty(\partial\mathbb{D})$$ be any function and consider the following subset of the closed unit ball of ${\mathcal H}^\infty:{\mathcal E}(\varphi)= \{k\in{\mathcal H}^\infty:\|k\|_\infty\leq 1\text{ and }\varphi-k\overline\varphi\in{\mathcal H}^\infty\}.$ Cowen’s theorem asserts that $$T_\varphi$$ is hyponormal if and only if $${\mathcal E}(\varphi)$$ is not empty.
In order to describe the set of functions $$g$$ such that $$T_{f+\overline g}$$ is hyponormal for a given $$f$$, C. C. Cowen [Pitman Res. Notes Math., Ser. 171, 155-167 (1988; Zbl 0677.47017)] introduced the following set: $G_f'= \Biggl\{g\in z{\mathcal H}^2: \sup_{h_0\in H}|\langle hh_0, f\rangle|\geq \sup_{h_0\in H}|\langle hh_0, g\rangle|\text{ for every }h\in{\mathcal H}^2\Biggr\},$ where $$H= \{h\in z{\mathcal H}^\infty:\|h\|_2\leq 1\}$$. If $$f\in{\mathcal H}^2$$, then there is a simpler expression for Cowen sets. In fact, $G_f'= \{g\in z{\mathcal H}^2: f+\overline g\in L^\infty(\partial\mathbb{D})\text{ and } T_{f+\overline g}\text{ is hyponormal}\}.$ In this form, it is called the reduced Cowen set.
In the paper under review, the authors adopt this more natural definition of Cowen set and, in addition, assume that $$f+\overline g\in L^\infty(\partial\mathbb{D})$$. To show the equivalence of the definitions of Cowen set for $$f\in {\mathcal H}^2$$, Hankel operators are needed. Recall that for $$\varphi\in L^\infty(\partial\mathbb{D})$$ the Hankel operator $$H_\varphi: H^2\rightarrow{\mathcal H}^2$$ is defined by $$H_\varphi f= J(I-P)(\varphi f)$$, where $$J:({\mathcal H}^2)^{\perp}\to{\mathcal H}^2$$ is defined by $$Jz^{-n}= z^{n-1}$$ for $$n\geq 1$$. For $$\varphi,\Phi\in L^\infty(\partial\mathbb{D})$$ the following relation between Toeplitz and Hankel operators holds: $$T_{\varphi\Phi}- T_\varphi T_\Phi= H^*_{\overline\varphi} H_\Phi$$.
It is easy to see that $$G_f'$$ is balanced and convex. Consider now $\nabla G_f'= \{g\in G_f': \lambda g\notin G_f'\text{ all }\lambda\in \mathbb{C},|\lambda|> 1\}$ and let $$\text{ext }G_f'$$ denote the set of extreme points of $$G_f'$$. In 1988 Cowen posed the following interesting question: Is $$\nabla G_f'$$ contained in $$\text{ext }G_f'$$? In an unpublished work M. Cho and the authors had previously given an affirmative answer to the above question in the case where $$f$$ is an analytic polynomial. Quite surprisingly, the authors of the present paper answer the above question in the negative. They show that if $$\Phi\in{\mathcal H}^\infty$$ and $$\overline\Phi$$ is not of bounded type and $$f(z)= z^3\Phi(z)$$, then $$\nabla G_f'$$ is not contained in $$\text{ext }G_f'$$. The proof is based on a functional equation that involves Hankel operators. As a more concete example, the authors provide the following one: if $$\Phi$$ is the Riemann mapping of the unit disk onto the interior of the ellipse with vertices in $$\pm i(1-\alpha)^{-1}$$ and passing through $$\pm(1+\alpha)^{-1}$$, where $$0< \alpha< 1$$, then $$\Phi$$ is in $${\mathcal H}^\infty$$ and $$\overline\Phi$$ is not of bounded type [C. C. Cowen and J. J. Long, jun. J. Reine Angew. Math. 351, 216-220 (1984; Zbl 0532.47019)]. Further, the authors give a class of functions more general than the analytic polynomials for which the answer to Cowen’s question is affirmative. They prove that if $$\text{rank }H_{\overline f}$$ is finite, then $$\nabla G_f'$$ is contained in $$\text{ext }G_f'$$. The proof of this result involves some deep techniques involving Blaschke products and the fact that these products are extreme points of the unit ball of $${\mathcal H}^\infty$$.
##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47B20 Subnormal operators, hyponormal operators, etc. 30D50 Blaschke products, etc. (MSC2000) 47B38 Linear operators on function spaces (general)
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