Bergman空间上以函数${\phi }_1\left(r\right){\text{e}}^{ip\theta }+{\phi }_2\left(r\right){\text{e}}^{iq\theta }$为符号的Toeplitz算子的亚正规性

Hyponormality of Toeplitz Operators with Symbols

摘要: 本文主要研究单位圆盘的Bergman空间上带有非调和符号的Toeplitz算子的亚正规性，得到以函数$\phi \left(z\right)=r^m{\text{e}}^{i\delta \theta }$ 为符号的Toeplitz算子是亚正规的当且仅当$\delta \ge \text{0}$ ，以函数$\phi \left(z\right)=a{r}^m{\text{e}}^{i\delta \theta }+b{r}^m{\text{e}}^{-i\delta \theta }$ ($\delta >\text{0}$ )为符号的Toeplitz算子是亚正规算子的充要条件是$\left|a\right|\ge \left|b\right|$ ，且给出以函数$\phi \left(z\right)=a{r}^m{\text{e}}^{i\delta \theta }+b{r}^n{\text{e}}^{-i\delta \theta }$ ($m\ne n$ )为符号的Toeplitz算子是亚正规算子的必要条件。此外，还得到Toeplitz算子$T_{\phi }$ 是亚正规的当且仅当$T_{a\phi }$ ($a\ne 0$ )是亚正规的，当且仅当对任意的复数$a,b$ ，$T_{a\phi +b}$ 是亚正规的。

Abstract: This paper investigates the hyponormality of Toeplitz operators with non-harmonic symbols on the Bergman space over the unit disk. It is proved that the Toeplitz operator with symbol $\phi \left(z\right)=r^m{\text{e}}^{i\delta \theta }$ is hyponormal if and only if $\delta \ge \text{0}$ . For the Toeplitz operator with symbol $\phi \left(z\right)=a{r}^m{\text{e}}^{i\delta \theta }+b{r}^m{\text{e}}^{-i\delta \theta }$ ($\delta >\text{0}$ ), we show that this operator is hyponormal if and only if $\left|a\right|\ge \left|b\right|$ , and necessary conditions for hyponormality are established for Toeplitz operators with symbol $\phi \left(z\right)=a{r}^m{\text{e}}^{i\delta \theta }+b{r}^n{\text{e}}^{-i\delta \theta }$ ($m\ne n$ ). In addition, we prove that the Toeplitz operator $T_{\phi }$ is hyponormal if and only if $T_{a\phi }$ ($a\ne 0$ ) is hyponormal, which is also equivalent to the hyponormality of $T_{a\phi +b}$ for arbitrary complex scalars $a,b$ .