 Invertibility from lower bounds on an operator and its adjoint

A bounded operator $T$ on a Hilbert space is invertible exactly when both $T$ and $T^{\ast }$ are bounded below. A lower bound on $T$ makes it injective with closed range, while a lower bound on $T^{\ast }$ gives $\ker T^{\ast }=0$ and hence dense range throughClosed dense range is the whole space, and the lower bound controls the inverse.