Let $X$ be a Banach space and $T:X\to X$ a continuous linear isomorphism. We say that $T$ is an hyperbolic isomorphism if its spectrum is disjoint with the unit circle, i.e. $\sigma(T)\cap \{z\in \C:|z|=1\}=\emptyset$ .

If this is the case, by the spectral theorem there is a splitting of $X$ into two invariant subspaces, $X=E^s\oplus E^u$ (and therefore, a corresponding splitting of $T$ into two operators $T^s:E^s\to E^s$ and $T_u:E^u\to E^u$ , i.e. $T=T_s\oplus T_u$ ), such that $\sigma(T_s) = \sigma(T)\cap \{z:|z|<1\}$ and $\sigma(T_u)=\sigma(T)\cap\{z:|z|>1\}$ . Also, for any $\lambda$ greater than the spectral radius of both $T_s$ and $T_u^{-1}$ there exists an equivalent (box-type) norm $\|\cdot\|_1$ such that $$\|T_s\|_1 < \lambda \textnormal{ and } \|T_u^{-1}\|_1 < \lambda$$ and $$\|x\|_1 = \max\{\|x_u\|_1,\|x_s\|_1\}.$$ In particular, $\lambda$ can be chosen smaller than $1$ , so that $T_s$ and $T_u^{-1}$ are contractions.