Closed $ EP $ and Hypo-$ EP $ Operators on Hilbert Spaces

A bounded linear operator $A$ on a Hilbert space $\mathcal H$ is said to be an $EP$ (hypo-$EP$) operator if ranges of $A$ and $A^*$ are equal (range of $A$ is contained in range of $A^*$) and $A$ has a closed range. In this paper, we define $EP$ and hypo-$EP$ operators for densely defined closed linear operators on Hilbert spaces and extend results from bounded operator settings to (possibly unbounded) closed operator settings.