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Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols. (English) Zbl 1318.47031
In [Pac. J. Math. 223, No. 1, 95–111 (2006; Zbl 1125.47019)], C.-X. Gu et al. extended Cowen’s theorem to block Toeplitz operators: For each $$\Phi \in L^\infty_{M_n}$$, let $$\mathcal{E}(\Phi):= \{ K \in H^\infty_{M_n} : \|K\|_\infty \leq 1 ~\text{and}~ \Phi- K\Phi^* \in H^\infty_{M_n} \}$$. Then $$T_\Phi$$ is hyponormal if and only if $$\Phi$$ is normal and $$\mathcal{E}(\Phi)$$ is nonempty. First, the authors establish M. B. Abrahamse’s theorem [Duke Math. J. 43, 597–604 (1976; Zbl 0332.47017)] for matrix-valued rational symbols. Let $$\Phi \in L^\infty_{M_n}$$ be a matrix-valued rational function having a “matrix pole”, i.e., there exists $$\alpha \in \mathbb{D}$$ for which $$\ker H_\Phi \subseteq (z- \alpha)H^2_{\mathbb{C}^n}$$, where $$H_\Phi$$ denotes the Hankel operator with symbol $$\Phi$$. Then the authors prove that, if (i) $$T_\Phi$$ is hyponormal and (ii) $$\ker[T_\Phi^*, T_\Phi]$$ is invariant for $$T_\Phi$$, then $$T_\Phi$$ is normal. Hence, in particular, if $$T_\Phi$$ is subnormal, then $$T_\Phi$$ is normal. Next, the authors establish Amemiya-Ito-Wong’s theorem [I. Amemiya et al., Proc. Am. Math. Soc. 50, 254–258 (1975; Zbl 0339.47019)] for matrix-valued rational symbols. They prove that every pure quasinormal Toeplitz operator with a matrix-valued rational symbol is unitarily equivalent to an analytic Toeplitz operator.

##### MSC:
 47B20 Subnormal operators, hyponormal operators, etc. 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 15A83 Matrix completion problems 30H10 Hardy spaces 47A20 Dilations, extensions, compressions of linear operators
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