Let $B(H)$ denote the $C^*$-algebra of all bounded linear operators from a Hilbert space $H$ into itself. Let $T \in B(H)$. Define $L_T : B(H) \to B(H)$ by $L_T(S) = TS$ and define $R_T : B(H) \to B(H)$ by $R_T(S) = ST$. Consider the following three statements: (i) $T$ has closed range in $H$, (ii) $L_T$ has closed range in $B(H)$, (iii) $R_T$ has closed range in $B(H)$. It is proved that all these three statements are equivalent. Some possibilities of extending this result to Banach spaces have been discussed.