Let $A$ be a linear bounded operator acting on a Banach space $X.$ Assume the spectral radius of $A$ is equal $1.$ Do there exist invertible operators $U_n:X\to X,$ such that $$\|U_n^{-1}AU_n\|<1+{1\over n},\quad n\ge 1\ ?$$
I can do it for Hilbert space $X$ (see Szwarc, Ryszard; Problems and Solutions: Solutions of Advanced Problems: 6496. Amer. Math. Monthly 94 (1987), no. 2, 197–199). For the Banach space $X,$ I am able to construct a new norm $\|\cdot\|_n$ on $X,$ equivalent to the original norm $\|\cdot \|,$ such that the norm of $A:(X,\|\cdot\|_n)\to (X,\|\cdot\|_n)$ is less than $1+1/n.$ But I am unable to decide whether it can be done by similarity, as described above. I have checked (to be on the safe side) it works for two-dimensional space with norm $\|(z_1,z_2)\|=|z_1|+|z_2|.$ In general, the conclusion is true for any finite-dimensional normed space $X=\mathbb{C}^n,$ once it holds for the euclidean norm. This should follow from Jordan's decomposition of matrices and the fact that any two norms on finite-dimensional space $M_{n\times n}(\mathbb{C})$ are equivalent.
