Invertibility from lower bounds on an operator and its adjoint
= 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 $T^*$ are bounded below. A lower bound on $T$ makes it injective with closed range, while a lower bound on $T^*$ gives $\ker T^*=0$ and hence dense range through
$$
(\operatorname{ran}T)^\perp=\ker T^*.
$$
Closed dense range is the whole space, and the lower bound controls the inverse.