The Hyponormal Toeplitz operators on the vector valued Bergman space

Yufeng Lu; Puyu Cui; Yanyue Shi

doi:10.4134/BKMS.2014.51.1.237

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2014; 51(1): 237-252

Printed January 1, 2014

https://doi.org/10.4134/BKMS.2014.51.1.237

The Hyponormal Toeplitz operators on the vector valued Bergman space

Yufeng Lu, Puyu Cui, and Yanyue Shi

Dalian University of Technology, Dalian University of Technology, Ocean University of China

Abstract

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Abstract

In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators $T_{\Phi}$, where $\Phi = F + G^\ast$, $ F(z), \ G (z)$ are some matrix valued polynomials on the vector valued Bergman space $L^2_a (\mathbb{D},\mathbb{C}^n )$. We also show some necessary conditions for the hyponormality of $T_{F + G^\ast}$ with $F + G^\ast \in h^\infty \otimes M_{n \times n }$ on $L^2_a (\mathbb{D},\mathbb{C}^n )$.

Keywords: hyponormality, block Toeplitz operator, block Hankel operator

MSC numbers: 47B38