Let $X$ and $Y$ be Banach spaces and $T\in\mathcal{L}(X,Y)$ be a bounded linear operator from $X$ to $Y$.
If $T$ is surjective, then the open mapping theorem says that there is a positive $\delta$ such that $TB_1\supset\delta B_2$, where $B_1$ and $B_2$ are open unit balls in $X$ and $Y$ respectively.
My question is how is the $\delta$ related to the norm of $T$, which gives a (sharp) bound for the norm of the inverse of $T$ if $T$ is also injective.
Thanks!
And another related question: If $\mu$ is at a positive distance to $\sigma(T)$, the spectrum of $T$, how is $\|(\mu-T)^{-1}\|$ related to the distance from $\mu$ to $\sigma(T)$? Obviously we have an lower bound, but what I need is an upper bound.
Thanks!