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.)