Reduced Cowen sets.

*(English)*Zbl 0990.47027Let \(\mathbb{D}\) 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 \({\mathcal H}^\infty\).

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 \({\mathcal H}^\infty\).

Reviewer: Alfonso Montes-Rodriguez

##### 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) |