ACTA issues

## Toeplitz type operators on the derivative Hardy space $S^2(\mathbb {D})$

 Abstract. A Toeplitz type operator $T_\phi$ with co-analytic symbol $\phi$ which can be seen as the adjoint of the multiplication operator on $S^2(\mathbb {D})$ is introduced and studied on the derivative Hardy space $S^2(\mathbb {D})$. The characterizations for the operator $T_\phi$ to be normal, self-adjoint and isometric on $S^2(\mathbb {D})$ have been obtained. In addition, it has been shown that the operator $T_{\bar {z}^k}$ for a fixed non-negative integer $k$ is a Fredholm operator and its point spectrum is the closed unit disk. DOI: 10.14232/actasm-018-805-0 AMS Subject Classification (1991): 47B35; 47B32 Keyword(s): Toeplitz operator, multiplication operator, derivative Hardy space Received June 9, 2018 and in final form July 27, 2018. (Registered under 55/2018.)