Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral radius of $T$, and suppose $r(T)<\lVert T\rVert$. I would like to renorm $X$ with an equivalent norm $|\cdot|$ so that $r(T)=|T|$, i.e. so that the spectral radius attains the operator norm.
It is already known that we can equivalently renorm $X$ so that the spectral radius almost attains the norm. Let $\epsilon>0$ be arbitrary and let $m$ be the smallest integer so that $\lVert T^m\rVert^{1/m}<r(T)+\epsilon$. Then we can define the equivalent norm $|\cdot|_{T,\epsilon}$ on $X$ by
$\displaystyle |x|_{T,\epsilon}:=\left[\lVert x\rVert^2+\left(\frac{\lVert Tx\rVert}{r(T)+\epsilon}\right)^2+\left(\frac{\lVert T^2x\rVert}{[r(T)+\epsilon]^2}\right)^2+\cdots+\left(\frac{\lVert T^mx\rVert}{[r(T)+\epsilon]^m}\right)^2\right]^{1/2}$
In this case, $|T|_{T,\epsilon}<r(T)+\epsilon$. However, this is not good enough for my purposes. We need equality, not almost-equality.
In general, what I want need not be possible, for instance if $T$ is a nonzero quasinilpotent operator. However, this problem arises in a context where I can make some powerful assumptions. In particular, I have a situation where $0<r(T)$ with $\sigma(T)$ uncountable, $0\in\partial\sigma(T)$, and $\partial\sigma(T)\subseteq\sigma_p(T)$, where $\sigma(T)$ denotes the spectrum and $\sigma_p(T)$ the point spectrum, and $\partial$ denotes the topological boundary. I can also assume that $\sigma_p(T^*)=\emptyset$, where $T^*\in\mathcal{L}(X^*)$ is the continuous dual of $T$.
At least sometimes, what I want really is possible. For instance let $(\delta_n)$ be a strictly decreasing sequence of positive reals with $\prod_{n=1}^\infty(1+\delta_n)<\infty$, and let $T\in\mathcal{L}(c_0)$ be the weighted left-shift operator defined by $Te_1=0$ and $Te_{n+1}=(1+\delta_n)e_n$, where $(e_n)$ is the unit vector basis for $c_0$, and $c_0$ (as usual) is the space of scalar sequences tending to zero equipped with the sup norm. Then it's easy to see that $1=r(T)<1+\delta_1=\lVert T\rVert$. Now, let $U\in\mathcal{L}(c_0)$ be the isomorphism $Ue_n=(1+\delta_n)e_n$, and define $|\cdot|$ by $|x|:=\lVert U^{-1}x\rVert$. Then $|\cdot|$ is an equivalent norm with $r(T)=|T|=1$.
Has this problem been studied at all?
Please note, I am also free to "adjust" $X$ by modding out finite-dimensional subspaces. More precisely, if $F$ is a finite-dimensional subspace of $X$ then I can replace $X$ with $W\cong X/F$, where $X=W\oplus F$ and $T$ with the corresponding induced operator $P_WT|_W$ (where $P_W$ is the continuous linear projection onto $W$ along $F$).