an:06284983
Zbl 1318.47031
Curto, Ra??l E.; Hwang, In Sung; Kang, Dong-O; Lee, Woo Young
Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols
EN
Adv. Math. 255, 562-585 (2014).
00331646
2014
j
47B20 47B35 46J15 15A83 30H10 47A20
Amemiya, Ito and Wong's theorem; subnormal; quasinormal; hyponormal
In [Pac. J. Math. 223, No. 1, 95--111 (2006; Zbl 1125.47019)], \textit{C.-X.\thinspace 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 \textit{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 [\textit{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.
Takanori Yamamoto (Sapporo)
Zbl 1125.47019; Zbl 0332.47017; Zbl 0339.47019