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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].

##### 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
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