Nice operators and nice spaces

Abstract

In this paper we obtain a necessary and sufficient condition for an operator on a uniform algebra to be nice. We characterize nice operators on an expansive class of Banach spaces. Then as examples, we give a complete description of nice operators on the space of differentiable functions on compact perfect plane sets, Bloch and Zygmund spaces and the space of Lipschitz functions on intervals. We also prove that every surjective nice operators are isometries on some of these Banach spaces.