Strict operator positivity does not imply surjectivity
= Strict operator positivity does not imply surjectivity
On $\ell^2$, the bounded <self-adjoint operator> $(Lv)_j=v_j/j$ is strictly positive. Yet $f_j=1/j$ lies in $\ell^2$ and $Lu=f$ would require the non-square-summable vector $u_j=1$. A positive uniform lower bound, rather than strict positivity alone, guarantees a bounded inverse for a bounded <self-adjoint operator>.