Operators with compatible ranges

A bounded operator $T$ on a finite or infinite--dimensional Hilbert space is called a disjoint range (DR) operator if $R(T)\cap R(T^*)=\{0\}$, where $T^*$ stands for the adjoint of $T$, while $R(\cdot)$ 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 $R(T)\cap R(T^*)$. We extend and improve some results about DR operators and derive some new results regarding the CoR class.