Let $A$ be a right invertible operator acting on a Hilbert space $\mathcal{H} $. I am trying to find a right inverse of $A$ that has the minimal spectal radius possible, that is an operator $B$ such that $AB=I$ and $\lim \left\Vert B^{n}\right\Vert ^{\frac{1}{n}}\leq \lim \left\Vert C^{n}\right\Vert ^{\frac{% 1}{n}}$ for every right inverse $C$ of $A$. I am suspecting that $B$ is determined by $Im\left( B\right) =\left(ker\left( A\right) \right) ^{\bot }$ but all I can prove in this case is that $\left\Vert B\right\Vert $ is minimal.
Thank you !