Differences of Stević–Sharma operators

Abstract

A generalization of the products of composition, multiplication and differentiation operators is the Stevic–Sharma operator $$T_{u_1,u_2,\varphi }$$ , defined by $$T_{u_1,u_2,\varphi }f=u_1\cdot f\circ \varphi +u_2\cdot f'\circ \varphi $$ , where $$u_1,u_2,\varphi $$ are holomorphic functions on the unit disk $${\mathbb {D}}$$ in the complex plane $${\mathbb {C}}$$ and $$\varphi ({\mathbb {D}})\subset {\mathbb {D}}$$ . We are interested in the difference of Stevic–Sharma operators which has never been considered so far. In this paper, we characterize its boundedness, compactness and order boundedness between Banach spaces of holomorphic functions. As an important special case, we obtain the above characterizations of the difference of weighted composition operators. Furthermore, we show the equivalence of order boundedness and Hilbert-Schmidtness for the difference of composition operators between Hardy or weighted Bergman spaces.