For two given Hilbert spaces $\mathcal H$ and $\mathcal K$ and a given bounded linear operator $A\in\mathcal L(\mathcal H,\mathcal K)$ having closed range, it is well known that the Moore-Penrose inverse of $A$ is a reflexive $g$-inverse $G\in\mathcal L(\mathcal K,\mathcal H)$ of $A$ which is both minimum norm and least squares. In this paper, weaker equivalent conditions for an operator $G$ to be the Moore--Penrose inverse of $A$ are investigated in terms of normal,$EP$, bi-normal, bi-$EP$, $\ell$-quasi-normal and $r$-quasi-normal and $\ell$-quasi-$EP$ and $r$-quasi-$EP$ operators.