A bounded operator T on a finite or infinite–dimensional Hilbert space is called a disjoint range (DR) operator if R(T) ∩ R(T *) = {0}, where T * stands for the adjoint of T, while R(·) denotes the range of an operator. Such operators (matrices) were introduced and systematically studied by Baksalary and Trenkler, and later by Deng et al. In this paper we introduce a wider class of operators: we say that T is a compatible range (CoR) operator if T and T * coincide on R(T) ∩ R(T *). We extend and improve some results about DR operators and derive some new results regarding the CoR class.