Subnormal operators of Hardy type.

*(English)*Zbl 0896.47019
Janas, Jan (ed.) et al., Linear operators. Proceedings of the semester organized at the Stefan Banach International Mathematical Center, Warsaw, Poland, February 7–May 15, 1994. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 38, 315-324 (1997).

Let \(H\) be a Hilbert space. An operator \(S:H\rightarrow H\) is called subnormal if there is a Hilbert space \(K\) containing \(H\) and a normal operator \(N:K\rightarrow K\) such that \(H\subset N(H)\) and \(N| _H=S\). We consider the minimal normal extension \(N\) of \(S\) and write \(N=mne(S)\). The relation between the spectra of \(S\) and its minimal normal extension is given in the spectral inclusion theorem
\[
\partial\sigma(S)\subset \sigma(N)\subset\sigma(S).
\]
We also assume that \(\sigma(N)\subset\partial\sigma(S) \) and therefore equality holds in the first inclusion.

Let \(E\) be a flat unitary bundle. For \(1\leq p<\infty\) the Hardy space \(H^p[E]\) is the set of all holomorphic cross-sections \(f:\Omega\rightarrow E\) such that there is a harmonic majorant \(h:\Omega\rightarrow [0,+\infty)\) satisfying \[ \| f(\lambda)\|^p\leq h(\lambda), \lambda\in \Omega. \] We define \(\| f\|^p=(h(\lambda_0))^{1/p}\) in which \(h\) is the least majorant of \(| f| ^p\). For \(\varphi\in H^{\infty}{(\Omega)}\) the Toeplitz operator \(T_{\varphi}:H^p[E]\rightarrow H^p[E]\) is defined by \(f\mapsto \varphi f\). The bundle shift \(T_E=T_z\) where \((T_Ef)(\lambda)=\lambda f(\lambda),\lambda\in \Omega\).

M. B. Abrahamse and R. G. Douglas [Adv. Math. 19, 106-148 (1976; Zbl 0321.47019), Proc. Roy. Irish Acad. Sect. A 74, 135-141 (1974; Zbl 0302.47009)] have introduced a model for linear operators on function spaces. They prove that \(S\) is unitarily equivalent to \(N\oplus T_E\) where \(N\) is a normal operator and \(T_E\) is a bundle shift. If \(S\) is pure subnormal, that is, it has no nontrivial normal part, then \(S\cong T_E\) bundle shift of some flat unitary bundle over \(\Omega=\sigma(S)^{\circ}\). The latter operator acts as \(M_z\) on the Hardy space \(H^2[E]\).

Let \(\Omega\subset\mathbb{C}\) be a domain. A bounded linear operator \(S\) is said to be of Hardy type with respect to \(\Omega\) if \(S\) is pure subnormal and \(\sigma(S)\subset\bar{\Omega}\) and \(\sigma(N)\subset\partial \Omega\), hence \(\sigma(N)\subset \partial\sigma(S)\).

We say that \(\Omega\) is a Parreau-Widom type domain if any flat unitary bundle \(E\) satisfies \(H^{\infty}[E]\neq 0\). For such a domain satisfying certain conditions it is shown that \(R(\overline{\Omega})\) is pointwise boundedly dense in \(H^{\infty}(\Omega)\) where \(R(\overline{\Omega})\) is the uniform closure of the space of rational functions with poles of \(\overline{\Omega}\). Moreover, \(\nu\ll m\) for all measures \(\nu\) representing a point \(\lambda_0\) where \(m\) is the harmonic measure for \(\Omega\).

In the present paper the author uses the results of M. V. Samokhin [Mat. Sb. 182, No. 6, 892-910 (1991; Zbl 0761.30020) (Russian)] to simplify his earlier construction in K. Rudol [Integral Equations Oper. Theory 11, No. 3, 420-436 (1988; Zbl 0645.47021)].

For the entire collection see [Zbl 0863.00036].

Let \(E\) be a flat unitary bundle. For \(1\leq p<\infty\) the Hardy space \(H^p[E]\) is the set of all holomorphic cross-sections \(f:\Omega\rightarrow E\) such that there is a harmonic majorant \(h:\Omega\rightarrow [0,+\infty)\) satisfying \[ \| f(\lambda)\|^p\leq h(\lambda), \lambda\in \Omega. \] We define \(\| f\|^p=(h(\lambda_0))^{1/p}\) in which \(h\) is the least majorant of \(| f| ^p\). For \(\varphi\in H^{\infty}{(\Omega)}\) the Toeplitz operator \(T_{\varphi}:H^p[E]\rightarrow H^p[E]\) is defined by \(f\mapsto \varphi f\). The bundle shift \(T_E=T_z\) where \((T_Ef)(\lambda)=\lambda f(\lambda),\lambda\in \Omega\).

M. B. Abrahamse and R. G. Douglas [Adv. Math. 19, 106-148 (1976; Zbl 0321.47019), Proc. Roy. Irish Acad. Sect. A 74, 135-141 (1974; Zbl 0302.47009)] have introduced a model for linear operators on function spaces. They prove that \(S\) is unitarily equivalent to \(N\oplus T_E\) where \(N\) is a normal operator and \(T_E\) is a bundle shift. If \(S\) is pure subnormal, that is, it has no nontrivial normal part, then \(S\cong T_E\) bundle shift of some flat unitary bundle over \(\Omega=\sigma(S)^{\circ}\). The latter operator acts as \(M_z\) on the Hardy space \(H^2[E]\).

Let \(\Omega\subset\mathbb{C}\) be a domain. A bounded linear operator \(S\) is said to be of Hardy type with respect to \(\Omega\) if \(S\) is pure subnormal and \(\sigma(S)\subset\bar{\Omega}\) and \(\sigma(N)\subset\partial \Omega\), hence \(\sigma(N)\subset \partial\sigma(S)\).

We say that \(\Omega\) is a Parreau-Widom type domain if any flat unitary bundle \(E\) satisfies \(H^{\infty}[E]\neq 0\). For such a domain satisfying certain conditions it is shown that \(R(\overline{\Omega})\) is pointwise boundedly dense in \(H^{\infty}(\Omega)\) where \(R(\overline{\Omega})\) is the uniform closure of the space of rational functions with poles of \(\overline{\Omega}\). Moreover, \(\nu\ll m\) for all measures \(\nu\) representing a point \(\lambda_0\) where \(m\) is the harmonic measure for \(\Omega\).

In the present paper the author uses the results of M. V. Samokhin [Mat. Sb. 182, No. 6, 892-910 (1991; Zbl 0761.30020) (Russian)] to simplify his earlier construction in K. Rudol [Integral Equations Oper. Theory 11, No. 3, 420-436 (1988; Zbl 0645.47021)].

For the entire collection see [Zbl 0863.00036].

Reviewer: K.Seddighi (Shiraz)

##### MSC:

47B20 | Subnormal operators, hyponormal operators, etc. |

30D40 | Cluster sets, prime ends, boundary behavior |

47B38 | Linear operators on function spaces (general) |

30D55 | \(H^p\)-classes (MSC2000) |

47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |