The Hyponormal Block Dual Toeplitz Operators

Abstract:

      We consider the hyponormality of the block dual Toeplitz operator $S_{\Phi}$ acting on the orthogonal complement of the vector valued weighted Bergman space $\left(A_{\alpha}^{2}(\mathbb{D},\mathbb{C}^{n})\right)^{\bot}$. We provide a necessary and sufficient condition for the hyponormality of $S_{\Phi}$ with matrix valued bounded harmonic symbol. Next we give some applications of the function $g_{w,s}\in\left(A_{\alpha}^{2}(\mathbb{D})\right)^{\bot}$, which is similar to the reproducing kernel of the weighted Bergman space $A_{\alpha}^{2}(\mathbb{D})$. Finally, we characterize that the quasinormality of dual Toeplitz operator on $\left(A^{2}(\mathbb{D})\right)^{\bot}$.